X-ray Diffraction from Materials

$: X-ray Diffraction from Materials$
X-ray Diffraction from Materials

2008 Spring SemesterLecturer; Yang Mo KooLecturer; Yang Mo Koo

Monday and Wednesday 14:45~16:00

10. X-ray Diffuse Scattering

10.1 Thermal Diffuse Scattering due to Atomic Vibration10.2 Ordering and Diffuse Scattering due to Static Displacement

H kHomework

X-ray & AT Laboratory, GIFT, POSTECH

10. X-ray Diffuse ScatteringDiffraction of x-ray: periodic array of atomsDiffraction of x-ray: periodic array of atoms

Break periodicity by some reasons → diffraction peaks diffuse↓

Thermal vibration of atomsLong range ordering in the alloysLocal precipitation in the alloys

Thermal vibration of atoms

ittbtftdi lTh( ) ( )

aswrittenbecanatommofntdisplacemeTherkr kk

kk δ

0

0mm twcosAt,u +−⋅=∑

wave between difference phase the is and vector,positionaverage theiswherek

r

kδ

0m

(1) A transverse wave, (2) A longitudinal wave*Position in the average structure


*Position in the average structure are shown dotted


Ph di iPhonon dispersion curveBrillouin zone boundary

For a k vector- 1 longitudinal wave- 2 transverse waves

Long range ordering in the alloysLong range ordering in the alloys

Long range order parameter: Sg g p-random atomic arrangement: S=0-perfectly ordered atomic arrangement: S=1



Local precipitation in the alloys

atomsgneighbotintheindicates atoms gneighborin of parameter order range short -

:ordering Local

lmn lmn:α

difference sizeatomic to due distancec interatomi local - gg

ABlmn

lmn

r


10.1 Thermal Diffuse Scattering due to Atomic Vibration

Th Fi t S tt i E ti( ){ }

highesttheSincewhere

+=

−⋅=∑∑ nmnm n

meu iexpffI

urr

rrsπ2

0

The First Scattering Equation

btt ithfitatomic the detect not can one 10 of

order the is atom offrequency vibrationhighesttheSince where

12

+= mmm

,

.urr

Hz).10~ :ray-(x them between differencefrequency huge of

because scatteringray - xtheofpositon

18

( ){ } ( ){ } scstteringray - xthe of average time the detectonly can One

−⋅−⋅=∑∑ nmnmnmeu iexpiexpffI uusrrs ππ 22 00( ){ } ( ){ }

average. time the described is Here, ><

∑∑ nmnmnm n

meu ppff

Time average term of the above equation can be simplified asTime average term of the above equation can be simplified as

( ){ } ( ){ } ( ){ } ...iiexpiexp nmnmnm −−⋅+−⋅−=−⋅ 4412

21 2212 uusuusuus πππ !!


10.1 Thermal Diffuse Scattering due to Atomic Vibrationnt,displaceme smallFor

( ){ } ( ){ }

[ ]

nmnm iexpiexp uusuus

⎤⎡⎤⎡

⎥⎦⎤

⎢⎣⎡ −⋅−≅−⋅ ππ

11

2212 2

p

( ) ( ) ( )( )[ ]nmnm iiexpiexpiexp usususus ⋅⋅⎥⎦⎤

⎢⎣⎡ ⋅−⎥⎦

⎤⎢⎣⎡ ⋅−= ππππ 222

212

21 22

intensityscatteringthereduceequationtheofsidehandrighttheof terms twofirst The

( ) ( )22 2212

21

nm ii usus ⋅−=⋅−= ππM

M. denoted is factor Waller-Debye factor. Waller-Debye the called is whichygqg

22

( )( )[ ] ( )( ) ( )( )

as expanded be can term lexponentia third The

⋅⋅+⋅⋅+=⋅⋅ 222122122 iiiiiiexp usususususus ππππππ( )( )[ ] ( )( ) ( )( )

( )( )!

!

⋅⋅⋅+⋅⋅+

++

32231

222

22122

nm

nmnmnm

ii

iiiiiiexp

usus

usususususus

ππ

ππππππ

form. analytical as rewritten be can crystalatomic -mono of equation scattering the Then,



( ) ( ){ }0022 2 ifI M− ∑∑( ) ( ){ }

( )( ) ( )( ) ( )( )!!

usususususus

rrss

32

0022

223122

21221

2

iiiiii

iexpefI

nmnmnm

m nnm

Meu

−

⎭⎬⎫

⎩⎨⎧ ⋅⋅⋅+⋅⋅+⋅⋅+⋅⋅+×

−⋅= ∑∑

ππππππ

π

( ) ( ) ( ) ( )

( ) vibrationthermalthebyreduceintensitypeakBragg:

!!

ssss

22

321022

32

If

IIIIef

M

M

−

− ⋅⋅⋅++++=⎭⎩

( )( )

( )vector. wavephonon the of aid withcondition Bragg the

satisfied is which vector scattering at scattering diffuse thermal order first The :vibration thermaltheby reduceintensity peak Bragg:

sss

1

022

IIef M

( )

( ) . vector scattering at scattering diffuse thermal order third The :vectors. wavephonon the econsecutiv two of aid withcondition Bragg the

satisfiediswhichvectorscatteringatscatteringdiffusethermalorder secondThe:

ss

ss

3

2

I

I

scattering diffuse order second the of magnitude The 1%. than lessTypically peak. Bragg of that withcomapred smallvery isintensity scatteing diffuse thermal the of magnitude The

forneededisintensity scatteringdiffusethermalordersecondtheofnCalculatio .experiment scatteringray - xanalysisof the in account to in taken isintensity scatteing diffuse thermal order

first the Therefore, .scattering diffuse order first the of that of percentfew a also isintensity


.temperture athigh negligible not isintensity thosed because experiment etemperatur highygp

10.1 Thermal Diffuse Scattering due to Atomic VibrationCalculation of the Thermal Diffuse Scattering Intensity

3N:PhononofNumberzone Brillouin st-I

the in Confined :Phonon

cellunit Cubic Simple :Crystal

321 N NNN =

Calculation of the Thermal Diffuse Scattering Intensity

(N). WavesnlLongitudia - (2N), WavesTransverse -

3N :Phonon ofNumber321

( ) ( )∑ +⋅= twcosAt 0rkru δ

ntdisplacemeAtomic ( ) ( )

∑

∑

=

+−⋅= nn

mE

twcosAt,

21

kkk

k

u

rkru δ

energyKinetic

&

( )∑ ∑

∑

⎭⎬⎫

⎩⎨⎧

+−⋅=

=

nn

nnkin

twsinAm

m E

20

21

2

kkkk

k rk

u

δω ⎭⎩



hthfki tiA

∑∑∑=

==k j

j,kj,kn

nkin ANmmE3

1

222

41

21 ωu&

phonontheofenergy kinetic Average

Total energy of atom is the sum of the kinetic energy and the potential energy. The averageenergy of atom is twice of the average kinetic energy, so that

3

1

3

1

22

11

21

k jj,k

k jj,kj,ktot EANmE ω

⎪⎫⎪⎧

== ∑∑∑∑==

( ) 211

j,k

j,kj,k

j,k

.kTE

kT/expE ω

ω

≅

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

+=

e),temperatur (room etemperatur high the For

s.oscillatorharmonic anfor where hh

222 22

j,kj,k

j,kj,k Nm

kTNm

EA

ωω==


10.1 Thermal Diffuse Scattering due to Atomic VibrationThe First Order Thermal Diffuse Scattering (TDS)

( ) ( ){ } ( )( ) becomes equation the of therm oder first the of average time The

iiiexpefI nmm n

nmM ususrrss 0022

1 222 ⋅⋅−⋅= ∑∑− πππ

The First Order Thermal Diffuse Scattering (TDS)

( ){ } ( ) ( ){ }

( ){ } ( ) ( ) ( ){ }

Aif

cosiexpef

M

nmm n

nmM

m n

k

rrkAsrrsk

k

00220022

0020022

212

2212 −⋅⋅−⋅=

∑∑∑

∑∑∑− ππ

( ){ } ( ) ( ) ( ){ }component. phonon each along vector unit is where

cosAiexpef

j

nmjj,k

j,km n

nmM

e

rrkesrrs 00220022 2212 −⋅⋅−⋅= ∑∑∑− ππ

( )( ) ( ) ( )π2

21

2

22221 AefI

,/eexcos

jj,k

j,kM

-ixix

ess ⋅=

+=

∑−

becomesintensity diffuse thermal order first the ip,relationsh the the Using

( ) ( ){ } ( ) ( ){ }[ ]

( ) ( ) ( ) ( ){ }πππ

ππππ

22211

2222

002222

0000

/I/IAef

/iexp/iexp

jjkM

m nnmnm

j,

kskses

rrksrrks

−++⋅=

−⋅−+−⋅+×

∑

∑∑−

( ) ( ) ( ) ( ){ }πππ 22222 00 /I/IAef j

j,kj,k kskses ++∑



( ) thifithihA 2/I k±( )

( ) of aid withcondition Brrag the is vector the whichofintensity ndiffractio

theisfigure, the inshown As 0

2

2

/

/I

ks

ks

±

±

π

π

vector,scatteringaat vectorphonon the 2

./

sk± π

( )( )

( ) locatedis whenexists of case the for method same

the following calculate us Let

0

0

0

22

2

//I.I

/I

kskss

ks

±±

±

ππ

π

( )

( )atoccursonly ndiffractio The . from point reciprocal

closest the is which the ofvicinity the at0

2

hkl

ppp/ cbaHkss

H

*** +++± π( )

( ) , of values small for meaningful is Sincevalues. smallvery are where

atoccurs

0 321

321

321

2

2 hkl

p,p,p/Ip,p,p

ppp/

ks

cbaHks

±

+++=±

π

π

( ) ( )

constant. a as assumed be can22221

jj,k A es ⋅π



overnintegratioanbyreplavedbecanequationtheofkoversumthelargeveryisNSince

E dbdi/2hfii l

the for element volume a And is pointsdensity The zone. Brillouin first the

overnintegratioanby replavedbe canequationtheofkoversumthe large,very isNSince

.N

3

3

*

*a/ ⎟⎠⎞⎜

⎝⎛

becomesintensity diffuse order first the integral,

anasExpressed by expressedisvectors/2 the of pointterminal .dpdpdpdv πk/ 3212*ak =π

( ) ( )11( ) ( ) ( )

( ) ( ){ }∫

∑

⎟⎟⎞

⎜⎜⎛

++×

⋅= −j

jj,k

M

dpdpdpN/I/I

AefI

3

22221

22

221

21

ππ

π

aksks

ess

*

( ) ( ){ }

( ) ( )∑

∫

⋅=

⎟⎟⎠

⎜⎜⎝

−++×

−

−

jj,kM

st BZ

AefN

dpdpdp/I/I

22222

32131 00

221

22

π

ππ

es

aa

ksks*

( ) ( )∑

⋅= − jM

j

cosNkTefI 2

2222

14

2

π esss

get can one equation, the to term amplitude average the Inserting

( ) ∑j j,km

efI 21 ωss



( ) ( )∑

⋅

=

jM

jk,jkk

cosNkTf

./w

2222 4 ess

kVkV

π

equation; following theby scattering diffuse thermal

orderfirstthecalculatedcan oneThen, vector, ofvelosity theis If

( ) ( )∑= −

j j,k

jM

mNkTefI 22

221

4

Vks π

The Second Order Thermal Diffuse Scattering (TDS)

( ) position at maxima sharp have have will

functions The 0 22 /'/I kks ±± ππ

The Second Order Thermal Diffuse Scattering (TDS)

that such space reciprocal in

22'

hklHkks

s

=±±ππ

occursonly ndiffractio The' ***kk⎟⎞

⎜⎛

values. smallvery are where

321

32122p,p,p

ppphkl*** cbaHkks +++=⎟

⎠⎞

⎜⎝⎛ ±±

ππ



be pressedbecanTDSordersecondThe

( ) ( ){ }−⋅= ∑∑− π221 0022

2 rrss iexpefIm n

nmM

byexpressedbecanTDS order second The

( ) ( ) ( ) ( )

( ) ( )⎬⎫⎨⎧

−⋅⎟⎞

⎜⎛

⎬⎫

⎨⎧

−⋅⎟⎞

⎜⎛×

⋅⋅×∑∑

ππ

ππ

22

2212

21

0000

2222

rrkrrk

eses

'coscos

A Aj,k 'j,'k

'j'j,'kjj,k

( ) ( )( ) ( )

( ) ( ) ( ) ( ) ( )−++=

⎭⎬

⎩⎨ ⋅⎟

⎠⎜⎝⎭

⎬⎩⎨ ⋅⎟

⎠⎜⎝

×

∑∑−

ππ

ππ

21211

22

22

222222

rrrr

AAfI

coscos

M

nmnm

BAcosBAcos2cosAcosB relation, cosine the Using

( ) ( ) ( ) ( ) ( )

( ) ( )⎢⎣

⎡

⎭⎬⎫

⎩⎨⎧

−⋅⎟⎠⎞

⎜⎝⎛ −++

⎭⎬⎫

⎩⎨⎧

−⋅⎟⎠⎞

⎜⎝⎛ ++×

⋅⋅=

∑∑

∑∑

πππ

πππ

ππ

222

222

22

228

0000

222

rrk'ksrrk'ks

esess

iexpiexp

AAefI

nmnm

j,k 'j,'k'j'j,'kjj,k

M

( ) ( ) ⎥⎦

⎤

⎭⎬⎫

⎩⎨⎧

−⋅⎟⎠⎞

⎜⎝⎛ −−+

⎭⎬⎫

⎩⎨⎧

−⋅⎟⎠⎞

⎜⎝⎛ +−+

⎣ ⎭⎩ ⎠⎝⎭⎩ ⎠⎝

πππ

πππ

ππππ

222

222

2222

0000 rrk'ksrrk'ks iexpiexp nmnm

m n

( ) ( ) ( ) ( )

⎬⎫

⎨⎧

⎟⎞

⎜⎛

⎟⎞

⎜⎛

⎟⎞

⎜⎛

⎟⎞

⎜⎛

⋅⋅= ∑∑− ππ 2212

21

81 222222

k'kk'kk'kk'k

eses

IIII

A Aefj,k 'j,'k

'j'j,'kjj,kM


⎭⎬

⎩⎨ ⎟

⎠⎞

⎜⎝⎛ −−+⎟

⎠⎞

⎜⎝⎛ +−+⎟

⎠⎞

⎜⎝⎛ −++⎟

⎠⎞

⎜⎝⎛ ++×

ππππππππ 22222222 0000 ssss IIII


⎞⎛ kk

hklonethemorebewouldthereatoccursndiffractio Since N.becomes

space reciprocal all for nintegratio the and the ofvicinity the at exists 0

,

I

hkl

hkl

=±±

⎟⎠⎞

⎜⎝⎛ ±±

Hkks

Hkks22 ππ

( ) ( ) ( ) ( ) ( )figure. the in shown as condition Bragg thesatisify which

AAefNI

,

'j'j'kjjkM

hkl

∑∑ ∑ ⋅⋅= − esess 22222222 21211

22

ππ

ππ

( ) ( ) ( ) ( ) ( )

form,integraltoareashadedtheConverting ondition.Bragg thesatisfy whichpoint reciprocal hkl for all of summation the is figure the of area shaded The

AAefNIhkl j,k 'j,'k

'j'j,'kjj,k∑∑ ∑H

k

esess2 22

222

ππ

( ) ( ) ( ) ( ) ( )

,gggg

dvN A AefNIhkl

'j'j,'kOBZj j'

jj,kM ∑∫ ∑∑ ⋅⋅= −

H *aesess 3

22222222 2

212

21

21 ππ

( ) ( ) ( )

asrewrittenbecanequationThis Zone.Brillouinst1-overlappedthe denotes OBZwhere

dvω

cosω

cosm

TNkVefIhkl

OBZj j' ',j'k

'j

k,j

jM ∑∫ ∑∑⋅⋅

= −

H

esesss 2

2

2

24

2

220

222

28π

cell.unittheofvolum the isV wherehkl j j ,jkk,j



A l f th th l diff tt i i t it f C B ll i h b lAn example of the thermal diffuse scattering intensity of Cu-Be alloy is shown below. The first order TDS is quite big close to Bragg peaks both room temperature and hightemperature. However, the second order TDS intensity is only significant at high temperature.e pe a u e


10.1 Thermal Diffuse Scattering due to Atomic VibrationMeasurement of the Elastic Constants of the CrystalMeasurement of the Elastic Constants of the Crystal

rucc ∂== εσ

stress the of definition the From

opqr

qopqrqropqrop

c

xcc

∂== εσ

onacceleratithetimesmassthetoequalforceresultantthe set one elenmet, volume small For stiffness.elastic the are where

qp

ropqr

p

op

tu

xxuc

xσ

∂∂

=∂∂

∂=

∂

∂20

22

ρ

dbib tit iTh

on.acceleratithetimesmasstheto equal forceresultant

( ){ }tiexpAu −⋅= 000 ω

equation aldifferncti the into u Inserting

asexpressedbecan vibrtionatomic The

0

rk k

rqpopqr

f,, ff

AkkcA

⎞⎛

=

321

0ρω

, and as vector wavethe of cosines ldirectiona the define us Let 2

k

rorrqpopqr ADAffcAV

.V

≡=

⎟⎠

⎞⎜⎝

⎛=

02ρ

ω

Then asvelocity phase the defne and k



eq ationtheRe ritting( )

( ) 0

0

2332

222211

1331222

111

=+−−

=+−−

DAρVDADA

DADAρVDA

equation theRewritting

( )( ) 02

333322311 =−+−

cA

ρVDADADA

aswrittenbecanthesystemcubictheconsideroneIfoftcoefficien the the of tdeterminan the whensolution haveonly equation This

000000

121112

121211

cccccc

c.A opqrj as writtenbe canthesystem,cubic theconsider oneIf of

0000000000000

44

44

111212

cc

ccc

( ) ( ) ( )

00000

2222

44

44

ffffVfff

cbecomes equation the of tdeterminan the stiffenss,elastic this Using

( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )

022

22

1442

311324412134412

32441222

12

3442

211214412

13441221441222

32

2442

111

=−++++

+−+++++−++

VffcfcffccffccffccVffcfcffccffccffccVffcfc

ρρ

ρ



f ll ithSi[100]lt(1)lF 001 fff

( ) waveallongitudin:

equation above the from obtained be can solutionfollowingtheSince [100].alongspropageate wavea (1), example For

00

001

321112

321

==≠→=

===

AAAcV

,, f, ff

l ( )

( ) wavesetransevers:

and

waveallongitudin:

000

00

4422

321

≠≠→

≠→

AAAcVV

A, AAρ

Vl

( )

equationabovethefromobtainedbecansolutionfollowing the Since [110]. along spropageate wavea (2), example For

wavesetransevers:

02121

000

321

3214422

21

===

≠≠=→==

,, f, ff

A,, AAρ

VV tt

( ) ( ) waveallongitudin:

equationabovethefromobtainedbe can solutionfollowing

021

3214412112 ==→++= , A AA ccc

ρ Vl

( ) ( )

( ) waveetransevers::

waveetransevers:

001

021

2

321121121

≠==→=

=−=→−=

AAAcV

, AAAccρ

Vt

( ) waveetransevers: : 44 002 3212

≠==→= , AAAcρ

Vt



( ) ( ) waveallongitudin:

equation above the from obtained be can solution followingtheSince [111].alongspropageate wavea (3), example For

421

313131

2

321

==→++=

===

AAAcccV

,, f, ff

( ) ( )

( ) ( ) wavesetransevers:

waveallongitudin:

031

423

32144121122

321441211

21=++→+−==

==→++=

AAAcccρ

VV

AAA cccρ

V

tt

l

ρ

intensity;scatteringdiffuse thermal order first the of Mesurement

( ) ( )

intensity; scatteringdiffuse

2

2222

14 cos

mNkTefI

j j,k

jM esss

⋅= ∑−

ωπ

( ) ( ) ( ) ( )

Inserting

23

2

23

2

21

2

2

2 coscoscoscos

,ω

j

j,kk,j

eseseses

Vk

⋅+

⋅+

⋅=

⋅

=

∑

. vector wavenallongitudia theby dcontributeonly is P at TDS order st1- The

23

22

21

2 VVVj j,kV∑



( )2 100 V/V bTDSd1thIf1btibTh ( )

2

2 100

l

ll

V

V./V

c

Then, .calculatedbecanTDS,order1-themeasureoneIf 1becomesequation aboveThe

ρl=11c

QpointtheFor

( ) ( ) ( )tibthil l t dbbt i diWh

Q,pointthe For

VVcos

Vsin

ttl22

2

2

2

0010

10001021

++φφ

.calculated be also can Q, atintensity diffuse the of tmeasuremen the Fromequation. abovetheusingcalculatedbecan obtained,isWhen

ccc

12

4411

Mathematical deri ation is based on the contin m model hich is ass med thatMathematical derivation is based on the continuum model which is assumed that|k|<<d. Because of this reason, there is a large error for large |k|. A modified model proposed by Baden and Powell;

( )( )2

2/d

/dsinVV

kk

k ∞=


10.1 Thermal Diffuse Scattering due to Atomic VibrationDetermination of Inter atomic Force ConstantsDetermination of Inter-atomic Force Constants

Assume that the atomic vibrations are harmonic and that the restoring forces acting between atoms can be described as Hooke’s law, the force on atom l is proportional to the relative displacements of all the other atoms The components of force along the threerelative displacements of all the other atoms. The components of force along the three orthogonal axes x, y, and z can be written

( ) ( ) ( ) ( ) ( ) ( ) ( ){ }uunlDuunlDuunlDlF lll −+−+−=∑( ) ( ) ( ) ( ) ( ) ( ) ( ){ }( ) ( ) ( ) ( ) ( ) ( ) ( ){ }( ) ( ) ( ) ( ) ( ) ( ) ( ){ }

uunlDuunlDuunlDlF

uunlDuunlDuunlDlF

uunlDuunlDuunlDlF

lnlznzyzlynyyylxnxyxy

lnlznzxzlynyxylxnxxxx

++

−+−+−=

++

∑

∑

∑

≠

≠

( ) ( ) ( ) ( ) ( ) ( ) ( ){ }

.atomexceptatoms all overextended sum the whereand n, atom of ntdisplaceme the of component th the represents where

lk-u

uunlDuunlDuunlDlF

nk

lnlznzzzlynyzylxnxzxz −+−+−=∑

≠

( )

directiontheinntdisplaceme relative unit a given is atom whendirection the in atom on force the determine

, atom and atom between acting constants forcec interatomi the quantities Thep

j-ni-l

ln,nlD ij

( ) ( )

atom whendirection th the in atom on force the gdeterminin constants as considered

bemay This . and where constants, additional gIntorducindirection. the in ntdisplaceme

li-l

zx, y,i, j,nlDllDj-

lnijij =−= ∑

≠


direction. the in displaced is itself j-


thenonacceleratiitsandmassitsofproductthetoatomanonforcetheEquating

( ) ( ) ( ) ( ){ }2

2

++== ∑n

iziziyiyixixli

li unlDunlDunlDdt

udmlF

thenon,acceleratiitsandmassitsofproductthetoatomanon force theEquating

( ) ( )0 +−⋅=∑ mm ,twcosAt,u rkr kkk

k

btithi

nt,displacemeatomic gindroducin to and cell, primitive per atom one withlatticesonly gconsiderin to restricts One

δ

( ) ( ) ( ) ( ){ } ( )002 +−⋅++=+−⋅ ∑ nn

z,kjzy,kjyx,kjxlj,kk twcosAnlDAnlDAnlDtwcosAm rkrk kkkk

symmetriccentroiscrystalthethatassumingtionsimplificaFor

- becomes equation this

δδω

( ) ( ) ( ){ } ( ){ }

( ) ( ) ( ) ( ){ }1 00

002 −⋅++=

∑

∑ nln

z,kjzy,kjyx,kjxj,kk

klDG

cosAnlDAnlDAnlDAm

k

rrk

i blthI t d i

- symmetric-centroiscrystalthethatassuming tion,simplifica For

ω

( ) ( ) ( ) ( ){ }( ){ } ( ) ( )( ) ( ){ } ( ) 0

0

1

2

2

00

=++−

−⋅−= ∑

k,zxzk,yxyk,xkxx

nnlijij

AGAGAG

AGAGAωG

rrkcosnlDmG

kkk

kkk

k

,variables,new thegIntroducin

( ) ( ){ } ( )( ) ( ) ( ){ } 0

02

2

=−++

=+−+

k,zkzzk,yyzk,xzx

k,zyzk,ykyyk,xyx

AωGAGAG

AGAωGAG

kkk

kkk



( ) ( ) ( )( ) ( ) ( )

vanish; must tscoefficien the of tdeterminan the solution, trivial non have To

=−

−

GωGG

GGωG

k

xzxykxx

2

2

0kkk

kkk

( ) ( ) ( )( ) ( ) ( )

tthiithelastic the fo vibrations of modes three the of sfrequencie the give solutions The

−ωGGG

GωGG

kzzyzzx

yzkyyyx

2

0

kkk

kkk

L tdtl ttithtti ltll lorient z and y, x,axes orthgonal three The etc. Cu Al,as such crystal FCC example, an As

vector. wavethis withwaves

( )

Let and vectorslatticethetoy repectiveltoparallel

++=

++=−

kkk

nnn

.,,

*z

*y

*x

ln

321

3322112100

321

aaak

aaarr

aaa

( ) ( ) ( )

equations, these Using

++−= ∑ knknkncosn,n,nDm

kG zyxn,n,n

ij 321321321

1

( ) ( ) ( ) ( ) ( ) ( ).101 110 011 011 101 110 be wouldneighbor nearest st1- the

for neighbors. its all and atomany over extends over sum The

±±±±±± ,,,,,

n,n,nln,n,n 321321



Consider the restrictions that the symmetry of FCC imposes on the interatomic force constants.

( ) ( )axis;ztoarprepediculplaneReflection

symmstry of Center

321321 = n,n,nD,n,nn Dxy-mirror plane

a produce not can plane reflection within lies whichntdisplaceme a 110, atom For

axis;ztoarprepedicul plane Reflection

( ) ( ){ }producenotcandirectionzofntdisplacemeThe

.000atomonforce of component-z

0110110 == zyzx DD

( ) ( ){ }. plane relection the in is whichforce of component a 000 atom on

producenotcandirectionzof ntdisplacemeThe

0110110 == yzxz DD


10.1 Thermal Diffuse Scattering due to Atomic VibrationgetcanoneatomsneibourotherthetosymmetryreflectionApplying

( ) ( ) ( ) ( )( ) ( ) ( ) ( )( ) ( ) ( ) ( )

getcanone, atomsneibourotherthetosymmetry reflectionApplying

====

====

0101101101101

0110110110110

yzyxyzxy

zyzxyzxz

DDDD

DDDD

( ) ( ) ( ) ( )( ) ( ) ( ) ( )( ) ( ) ( ) ( )

====

====

====

0110110110110

0011011011011

0011011011011

zyzxyzxz

xzxyzxyx

DDDD

DDDD

DDDD

( ) ( ) ( ) ( )( ) ( ) ( ) ( )

==== 0101101101101

0110110110110

xzxyzxyx

yzyxyzxy

DDDD

DDDD

thein000atomon forcetheexample,For symmetry. rotationalsame thisshow must 011 and 10,1 0,11 110, atoms, of ntsdisplaceme to due 000

atomonforcetheandsysmmetry rotational fold-4of axis an isaxis-z The

symmetryfold-4thesatisifyTodirectiony-thein011atomofntdisplaceme equal an to due direction-x the in 000 atom on force the to equal

be must direction- x the in 110 atom of ntdisplaceme a to due direction-ypy y

+++

exist must relations following thesymmetry,fold-4thesatisify To direction.y -thein011atom of ntdisplaceme


10.1 Thermal Diffuse Scattering due to Atomic Vibration( ) ( ) ( ) ( )DDDD 011110011110 ==( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( )

DD,D D

DD, DD, DD

DD,D D

xzzxzxxz

zzzzyyyyyyxx

yxxyxyyx

110101110101

011110011110011110

011110011110

−=−=

===

−=−=

( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( )

( ) ( ) ( )

D D,D D

DD, DD, DD,

xzzxzxxz

zzzzyyyyyyxx

xzzxzxxz

101011101011

110101110101110101

−=−=

===

( ) ( ) ( ) ( ) ( ) ( )

must110atomntdisplacemetodue000atomoncomponentsforcetheSince0].1[1 the to larperpendicu is whichplane reflection a in lie 110 and 000 Atoms

DD, DD, DD zzzzyyyyyyxx 101011101011101011 ===

( ) ( ) ( ) ( ) , that follows it plane, mirror {110} all for Similary, symmetry. thisshow

must110,atomntdisplacemetodue000,atomoncomponents forcetheSince

DD, DD yxxyyyxx 110110110110 ==

( ) ( ) ( ) ( )( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( )

. ,

DDDDDD

DD, DD DD, DD

zyyzzzyy

xzzxxxzz

101110101110101110

011011011011101101101101

===

====

( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( )( ) ( )

. DD

, DD, DD, DD

,DD,DD, DD

xxzz

yzyxzzzzxyxz

yyzzxxxxyzxy

011110

011110011101011110

101110101110101110

=

===

===


( ) ( )xxzz


( ) ( ) ( ) :tscoefficien threeonly by specified be can neighbors nearest twelve its

andatomanbetweenforcerestoringtheFCCinnsrestrictiosymmetry ofresulta As

111 101101101 zzyyxx D,D,αD γβ ===( ) ( ) ( )

dbi hbtdift tfThdi t more for constant force the to ionsconsideratsymmetry same theapply can One

111 zzyyxx ,, γβ

( ) ( ) ts;coefficien twoby

expressedbecanneighborsnearestsecondsix forconstants force The distance.

22 200200 yyxx .D,D βα ==( ) ( )

( ) ( )ts;coefficien fourby

expressed be can neighbors nearest third 24 for constants force The22

211211

yyxx

DD β( ) ( )( ) ( )

theingivenareconstantsforceneaighborthirdand second,first,The.

33

33

211211

211211

xyyz

yyxx

D,D

,D,D

δγ

βα

==

==

.negligible be to assumed be willneighbors third the beyond atoms all withnsinteractio , to lproportina is atoms the ofenergy binding the Since Table.

gg,,6-~r



( )( )

( )( )[ ] ( )

( ) ( )kiiicoscoscoscoscos

G

G

kkkkk

kkkkk

ij

kjkji

kjkji

⎪⎬

⎫⎪⎨

⎧ −++−

table.thusingcalculatedbecan quantities The

2222212221

1212

4 ββα

k

k

( ) ( ) ( )( )[ ]kcoscoscoskcoscos

coscoskcossinsinsinm

G

kkk

jk

kki

kkkii

jki

kjkji

⎪⎭

⎬⎪⎩

⎨+−+

−++++=

and

2223

22322

22

122

2

2212

βαβαk

( ) ( )( )ksinsinsinksincos

sinsinkcosm

Gj

kki

k

kkk

ij ijk

ji

⎥⎥⎦

⎤

⎢⎢⎣

⎡

+++

=

and

2223

2231

224

δγγ

k ( ).kj izx,y, i,j,k

ji

≠≠=⎥⎦⎢⎣

with, and where2223

obtained. becan sfrequencie theequation,secular theinto quantities theofon Substituti kω

direction;[100]in thegpropagatinwavesfor theequationthesolveletexample,For symmetry.high ofdirection afor simple quite become solutions theHowever, d.complicate

becan waveelastic arbittraryan for solution its and ,in equation cubic a is This 2ω

.allfor 0 and ,0direction;[100]in thegpropagatinwavesfor theequation thesolvelet example,For

j i Gkk ijzy ≠===



tdi litfd tj t thitilthftd t ithSi

( ) ( ) ( )sin24sin216equations; above theusing sfrequencie get threecan one

terms,diagonalitsofproduct just theisequation secular theoft determinantheSince

222 +++== ααβαω k xx kkG ( ) ( ) ( )

( ) ( ) ( )2

sin444

sin228

4sin2

4sin2

232

23211

22

32311

+++++==

+++==

βββαβαω

ααβαω

k

k

xxyy

xx

km

km

G

mmG

phonontheofcurvedispersionobtain thecanoneequation,thisUsing waves.e transversare and and wave,allongitudin a is where

321

23

22 =

ωωωωω

kli iB ill ifihd0di i[110]i hiF

direction. [100] alongphonontheof curvedispersionobtain thecan one equation,thisUsing

kkk( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) becomesequation secular theThen, . and

,0 all equation, G From . to

klimitszoneBrillouin first theand,,0direction,[110]in the gpropagatinFor

ij23

23

x

=

=====≤≤

==

kk

kkkkkk

yyxx

yxxyzyzxyzxzπ

xπ

yxz

GG

GGGGGGk-

kkk

( ) ( )( ) ( )[ ] ( ) ( )[ ] ( )[ ] 0

q,222 =−−−−+ ωωω kkkkk zzxyxxxyxx

yyxx

GGGGG


10.1 Thermal Diffuse Scattering due to Atomic Vibrationequation;thisofSolutions

( ) ( )

( ) ( )

equation; this ofSolutions

21

ksinksin

GG

xx

xyxx kk

⎢⎡ ++++++−+++=

+=

22422224 2313221

233311 γγββααδβαβα

ω

( ) ( )

( )

ksin

sinsinm

x⎥⎦⎤+++

⎢⎣+++++++++

43422

222

442222

2333

31322133311

δβα

γγββααδβαβα

( ) ( )

( ) ( )

22

ksinksin

GG

xx

xyxx kk

⎢⎣⎡ −−++++++++=

−=⎦

222

4422224 2

3132212

33311 γγββααδβαβα

ω

( ) ksin

m

x⎥⎦⎤−++

⎢⎣

43422

24

2333 δβα

( ) ( ) ( )

23

ksinksinksinm

G xxxzz k ⎥⎦

⎤⎢⎣⎡ +++++==

434

22

4444 2

32

3212

31 δαβββαω

( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) beconesequationseculartheThen

and equation, From tok limits zone Brillouin first the and direction, [111] the in gpropagatin For x

GGG

GGGGGGG.πk-π

,kkk

xzzyyzzxyzxyijx

zyx

kkk

kkkkkk =====≤≤

==


( ) ( ) ( ) becones equationseculartheThen, .GGG zzyyxx kkk ==


( ) ( )[ ] ( ) ( )[ ] ( ) ( )[ ]GGGGGG 02 222 kkkkkk( ) ( )[ ] ( ) ( )[ ] ( ) ( )[ ]

( ) ( ) equation; this of Solutions

21 xyxx

xyxxxyxxxyxx

GG

GGGGGG

2

02 222

+=

=−−−−−+

ω

ωωω

kk

kkkkkk

( )

( ) ] x

y

ksin

ksinm

4222

4242224

23333

231332211

++++

⎢⎣⎡ −++++++=

δγβα

γγβαβαβα

( ) ]( ) ( )

( )

23

22

x

xyxx

x

ksin

GG

ksin

2242224

422

231332311

3333

⎢⎣⎡ +−+++++=

−==

++++

γγβαβαβα

ωω

δγβα

kk

( )

( ) ] xksinm

2222

23333

31332311

−−++

⎢⎣δγβα

γγβββ

form simple the have,directionssymmtry high along gpropagatin wavesfor relations, dispersion The

xksinAmω 22 ∑=constants. forcec interatomi the of nscombinatio linear are where

n

nn

A

sinAmω41

∑=

=


10.1 Thermal Diffuse Scattering due to Atomic VibrationphaseawithtravelvectorwaveandfrequencyangularwithwavesElastic ω k

( )relaton obey the structure thisof directionssymmetry

three thealong gpropagatin waves theof s velocitiephase The ./Vvelocity phasea with travelvector waveandfrequency angular with wavesElastic

ωω

= kk

4sin1

12

2

2

k

Am

Vn

x

n= ∑= k

( )16

10

form the takes this0, h, wavelengtlong very oflimit In the

12

222 kAn

mV x

n=→

→

∑k

k

k

elastic ebetween th relations ofset a gives twoof comparisonA .elasticity theby determined s velocitieingcorrespond the toequalmust s velocitieThese

16 1m n= k

,81644 obtains one neglected, are

neighborsthirdbeyond nsinteractioIf constants.cinteratomi theand constants

332111 αααac β+++=

( )eparameterlattictheiswhere

.3284,204422

3314411

3321144

acc a

ααac δγγ

βββ++=+

++++=


.eparameterlattic theis where a

10.1 Thermal Diffuse Scattering due to Atomic VibrationCalculation of the Debye Waller FactorCalculation of the Debye-Waller Factor

( ) ( )AM jjk2

32

212 cos22

by expressed is atom of kind oneonly containingfactor Waller Debye

π∑∑=

−

es,s ( ) ( )

( )M

jk j

jk

23

22

1,2

11142

termre temperatuusingequation thisRewritting

π ∑∑

∑∑=

⎥⎤

⎢⎡

+h ess

,

( ) ( )kTmN

M jk j jkjk

2

1 ,,

etransverspureandallongitudinpurearewavesAlli) thatassumes one Debye, of method theFollowing

cos21/exp

2 ωω∑∑

= ⎥⎥⎦⎢

⎢⎣

+−

=h

es,s

( )k 3m

ihfd ih/ill ihfhl

is volume whose,k of radius of shpere a assumed is zoneBrillouin The iii) constant are waves theof veloctiesaverage The ii)

etransverspureandallongitudinpure are wavesAlli)

( )

( )

k

j

m

2

BZ

3BZ

obtaincan on equation, the to/sin2 and 1/3 is cos Inserting

.N/V is sphere in the pointstheofdensity The .3/4VzoneBrillouin theof that toequal

λθ

π

=

=

ses,( )

( ) dkkVN

mNM

BZj

k

jkmj

j

m 23

10

,3,

22

34

21

1exp11sin242 πωωλ

θπ ∑ ∫= ⎥

⎥⎦

⎤

⎢⎢⎣

⎡+

−⎟⎠⎞

⎜⎝⎛=

h

h


10.1 Thermal Diffuse Scattering due to Atomic VibrationFromconstantarewavesofvelocitiestheAssuming /ωV = k

( ) and

From constant.arewavesofvelocities theAssuming

dsin

./Vωπ/V/Vdωωdk k

,/ωV

jj,mBZjjj

jj

mj

⎥⎤

⎢⎡

⎟⎞

⎜⎛

==

=

∑ ∫h 322

33322

11124

34

θπ ω

k

( )Let

dkT/exp

sinmN

M jjj j,km,j

m,j

⎥⎥⎦⎢

⎢⎣

+−

⎟⎠⎞

⎜⎝⎛= ∑ ∫

= h

h

103 2

11

11242 ωωωωλ

θπ

( ) Debyeby introduced function the Using

dxΦ

,T/kTx,kTω

jm,j

jj Θ===

hh

1

ωε

( )

becomes 2M

e

dx

xΦx

−= ∫0 1

1 εεε

( )

thefortscoefficienandestemperatursticcharacteriseparateusLet

x

xxΦsin

mkTM

j

jj

j⎥⎦

⎤⎢⎣

⎡+

Θ⎟⎠⎞

⎜⎝⎛= ∑

=

h 3

12

222

41242

λθπ

. : wavestransverse the and allongitudin

thefortscoefficienandestemperaturstic characteri separateus Let

T/Θx,T/Θ x,k/ω Θ,k/ω Θ

x

ttllm,ltm,ll

j

==== hh


10.1 Thermal Diffuse Scattering due to Atomic VibrationrelationsthesegIntroducin

( ) ( )

relations thesegIntroducin

tt

t

ll

l

xxΦxxΦsinmk

TM⎭⎬⎫

⎩⎨⎧

⎥⎦⎤

⎢⎣⎡ +

Θ+⎥⎦

⎤⎢⎣⎡ +

Θ⎟⎠⎞

⎜⎝⎛=

42

41242 22

222

λθπ h

( )

that shownreadily isit term,lexponentia the expandingBy xxxxΦ ⋅⋅⋅+−+=⎥⎦

⎤⎢⎣⎡ +

3600361

4

42

( ) ( )[ ] ( ) ( )[ ]( ) ( ) ( )by defined factor,

2M the in use for average an introduceNow . 2, xFor M

tlM

ttll

iT

./Θ/Θ/Θ

/xxΦ/xxΦ

⎤⎡⎞⎛

+=

Θ+≈+<

212

213

44

222

222

θh ( )

great, not is however, ,difference The heat.specific the of the as same not is The

DM

M

ΘΘ

xxΦsinmkΘ

TM ⎥⎦⎤

⎢⎣⎡ +⎟

⎠⎞

⎜⎝⎛=

42122 2

22

λθπ h

constants.elastic threefrom2M for valueaobtain to possible also is It s.intensitie integrated the reduce to is vibrations thermal of effect The

. temperture Debye tabulated the for use can one ionapproxiamt first a to and DM ΘΘ


X-ray Diffraction from Materials

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