VIII. Kinematical Theory of Diffraction8-1. Total Scattering Amplitude

The path differencebetween beamsscattered from thevolume elementapart is

r

rkkrkrk )(2)ˆˆ( ''

The amplitude of the wave scattered from avolume element is proportional to the localelectron concentration ! Why?)(rn

kk

ˆ2

)ˆˆ( ' rkrk

Total scattering amplitude F

dVernrdernF rkki

V

rkki )()( ''

)()(

Define kkk'

dVernF rki )(

)(rn : periodic function in a crystalA periodic function can be expanded byFourier series!Any criteria to expand it?

Crystal translation vector)(rn )( Trn

Physical properties function of a crystal

Crystal: periodic

Periodic function Exponential Fourier Series

k

rkik enrn

2)( k

Tkirkik eenTrn

22)(

12 Tkie

cwbvauT

Translation vectors of the original crystal lattice

1 2 Tkie

for all

u, v, w: integer

*** clbkahk If

Vectors of the reciprocal latticeh, k, l: integer

T

)()( *** cwbvauclbkahTk

)( lwkvhu always integer

1 2 Tkie

Therefore, is not arbitrary! is reciprocallattice, i.e.

k

k

G

G

rGiGenrn

2)(

dVeendVernF rki

G

rGiG

rki

2)(

G

rkGiG

G

rkirGiG dVendVeen

)2(2

G

rkGiG dVenF

)2(

(i) when Gk

2

22 kk VndVnF

*

2222

kk nnVFI

(ii) when Gk

2

G

rkGiG

G

rkGiG dVendVenF

)2()2(

)2( Gk

G

G GknF

)2(

For an infinite crystal

;

Therefore, diffraction occurs at Gk

2

The total scattering amplitude F

dVerndVernF rGirki 2)()(

GrGi NSdVernN

cellunit

2)(

structure factor

For a unit cell, total electron concentration atdue to all atoms in the unit cellr

S

jjj rrnrn

1

)()(

cellunit

2)( dVernS rGiG

dVerrn rGiS

jjj

2

1

)(

S

j

rGijj dVerrn

1

2)(

S

j

rGirrGijj dVeerrn jj

1

2)(2)(

S

j

rrGijj

rGiG dVerrneS jj

1

)(22 )(

jf atomic form factor

S

jj

rGiG feS j

1

2

The scattering amplitude is then expressed as

S

jj

rGiG feNNSF j

1

2

8-2. Atomic form factor calculationThe meaning of form factor is equivalentto the total charge of an atom, which can beobtained by a direct calculation.

With the integral extended over the electronconcentration associated with a singleatom, set the origin at the atom

dVerndVerrnf rGij

rrGijjj

j 2)(2 )()(

Spherical integration dV = dr(rd) (rsind)

http://pleasemakeanote.blogspot.tw/2010/02/9-derivation-of-continuity-equation-in.html

r: 0 - : 0 - : 0 - 2

dVerndVerrnf rGij

rrGijjj

j 2)(2 )()(

0 0

2

0

2 )sin)(()(r

rGijj drrddrernf

rr

ddrrdrrddr

0

2

0

2

0

sin2)sin)((

3

34 r = 2

3

3r

0 0

2

0

cos22 sin)(r

iGrjj ddrderrnf

)()( rnrn jj cosGrrG

assume

rG

0 0

2

0

cos22 sin)(r

iGrjj ddrderrnf

0 0

cos22 sin)(2r

iGrjj dedrrrnf

0

cos2

0

cos2 )cos(sin dede iGriGr

0

cos2 )cos2(2

1 iGrdeiGr

iGr

iGree iGriGr

2

0cos2cos2

GrGr

)2sin(

0

2

2)2sin(2)(2

rjj Gr

Grdrrrnf

0

2

2)2sin()(4

rjj Gr

Grdrrrnf

depends on Gjfsin2* hklG

(chapter 7)

0When 0G 12

)2sin(lim0

Gr

Gr

zdrrrnfr

jj

0

2)(4 total number of

electron in an atom

0 Tabulatedjf Function of element and diffracted angle

Example:

Si, 400 diffraction peak, with Cu K (0.1542 nm)

nm 13577.04/54309.0400 d6.341542.0sin2 400 d

1

368.0sin

sin 0.2

9.67

526.7368.03.04.03.0

22.820.722.8

f

f

0 0.114.0 12.16

0.5 0.66.24 5.31

0.3 0.48.22 7.20

d21sin

or

8-3. Structure Factor Calculation

S

j

cwbvauclbkahij

S

j

rGijG

jjjj efefS1

)()(2

1

2 ***

S

j

lwkvhuijG

jjjefS1

)(2

(a) Primitive cell 1 atoms/unit cell; 000 and 100, 010, 001, 110, 101, 011, 111: equipoints of rank 1; Choose any one will have the same result!

ffeS lkhiG )000(2 for all hkl

(b) Base centered cell 2 atoms/unit cell; 000 and 100, 010, 001, 110, 101, 011, 111: equipoints of rank 1; ½ ½ 0, ½ ½ 1 : equipoints of rank 1; Two points chosen: 000 and ½ ½ 0.

S

jj

lwkvhuiG feS jjj

1

)(2

fSG 2

0GS

If h , k is unmixed or h + k is even

If h , k is mixed or h + k is odd

)021

21(2)000(2 lkhilkhi

G fefeS

]1[ )( khief

The meaning of the reflection condition(reflection rule)

Suppose that we have a square lattice

(a) Primitive unit cell： one atom at [00]

ffeS khiG )00(2

1

1

)(2

jj

kvhuiG feS jj

(b) Unit cell： two atoms at [00] [0]

1

1

)(2

jj

kvhuiG feS jj

fefefeS ihkhikhiG )1(

)021(2)00(2

fSG 2 If h is even0GS If h is odd

The reflection conditions remove the latticepoints [odd, 0] in the reciprocal lattice.

Therefore, the meaning of the reflection rule isto remove the additional lattice points due to theselection of unit cell. After the substraction, the reciprocal latticestructures derived from both cases (primitiveunit cell and unit cell) become the same.

(c) Body centered cell 2 atoms/unit cell; 000 and 100, 010, 001, 110, 101, 011, 111: equipoints of rank 1; ½ ½ ½: equipoints of rank 1; Two points to choose: 000 and ½ ½ ½.

S

j

lwkvhuijG

jjjefS1

)(2

fSG 2

0GS

If h + k + l is even

If h + k + l is odd

)21

21

21(2)000(2 lkhilkhi

G fefeS

]1[ )( lkhief

(d) Face centered cell 4 atoms/unit cell; 000 and 100, 010, 001, 110, 101, 011, 111: equipoints of rank 1; ½ ½ 0, ½ 0 ½, 0 ½ ½, ½ ½ 1, ½ 1 ½, 1 ½ ½: : equipoints of rank 3; Four points chosen: 000, ½½0, ½0½, 0½½

)21

210(2)

210

21(2

)021

21(2)000(2

lkhilkhi

lkhilkhiG

fefe

fefeS

]1[ )()()( lkilhikhiG eeefS

If h, k, l unmixed

If h, k, l mixed

fSG 4

0 GS

(e) close-packed hexagonal cell 2 atoms/unit cell 8 corner atoms: equipoints of rank 1; 2/3 1/3 1/2: equipoints of rank 1; Choose 000, 2/3 1/3 1/2. (Miller Indices)

)21

31

32(2)000(2 lkhilkhi

G fefeS

]1[)

232(2)

232(2 lkhilkhi

effef

]1][1[)

21

31

32(2)

21

31

32(222 lkhilkhi

G eefSI

]1][1[)

21

31

32(2)

21

31

32(22 lkhilkhi

eefI

]11[)

21

31

32(2)

21

31

32(22

lkhilkhieef

)]}21

31

32(2cos[22{2 lkhf

)]23

2([cos4 22 lkhf

2h + k l

3m3m

3m13m1

oddevenoddeven

)23

2(cos2 lkh

01

0.750.25

04f 2

3f 2

f 2

2

GS

(f) ZnS: four Zinc and four sulfur atoms per unit cell (8 atoms/unit cell)

000 and 100, 010, 001, 110, 101, 011, 111:equipoints of rank 1; (S atom)½½0, ½0½, 0½½, ½½1, ½1½, 1½½:equipoints of rank 3; (S atom)¼¼¼, ¾¾¼, ¾¼¾, ¼¾¾:equipoints of rank 4; (Zn)

Eight atoms chosen: 000, ½½0, ½0½,0½½ (the same as FCC), (Say S atom)¼¼¼, ¾¾¼, ¾¼¾, ¼¾¾ (Say Zn atom)!

)43

43

41(2)

43

41

43(2

)41

43

43(2)

41

41

41(2

)21

210(2)

210

21(2

)021

21(2)000(2

lkhi

Zn

lkhi

Zn

lkhi

Zn

lkhi

Zn

lkhi

S

lkhi

S

lkhi

Slkhi

SG

efef

efef

efef

efefS

]1[

][)()()(

)41

41

41(2)000(2

lkiS

lhikhi

lkhi

Znlkhi

SG

efee

efefS

Structure factor of face centered cell

fcc

lkhi

ZnSG SeffS

][)

21

21

21(

h, k, l: mixed 0fccS 0 GS

h, k, l: unmixed 4fccS

][4)

21

21

21( lkhi

ZnSG effS

when h, k, l are all odd ][4 ZnSG iffS

when h, k, l are all even and h + k + l = 4n ][4 ZnSG ffS

when h, k, l are all even and h + k + l 4n ][4 ZnSG ffS

]][[)

21

21

21()

21

21

21(22 lkhi

ZnS

lkhi

ZnSfccG effeffSS

]

[

)21

21

21(

)21

21

21(2222

lkhi

ZnS

lkhi

ZnSZnSfccG

eff

effffSS

)](2

cos2[ 2222lkhffffSS ZnSZnSfccG

)(16 222

ZnSG ffS

22)(16 ZnSG ffS

22)(16 ZnSG ffS

h+k+l : oddh+k+l :odd multiple of 2.h+k+l: even multiple of 2

Or

8-4. Shape EffectFor a finite crystal, assuming rectangularvolume

G

rGkiG dVenF

)2(

For a particular G

, if 02 DGk

dVenF rDiG

1

0

1

0

1

0

)(1

0

1

0

1

0

M

x

N

y

L

z

czbyaxDiG

M

x

N

y

L

z

rDiG enenF

1

0

1

0

1

0

M

x

N

y

L

z

czDibyDiaxDiG eeen

cDi

cDiL

bDi

bDiN

aDi

aDiM

G ee

ee

eenF

11

11

11

2/2/

2/2/

2/

2/

11

aDiaDi

aDiMaDiM

aDi

aDiM

aDi

aDiM

eeee

ee

ee

)2/sin(2)2/sin(2

2/

2/

aDiaDMi

ee

aDi

aDiM

)2/sin()2/sin(

)2/sin()2/sin(

)2/sin()2/sin(

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

cDcDL

bDbDN

aDaDM

eeenF cDLibDNiaDMiG

2FI 222

*

)2/sin()2/sin(

)2/sin()2/sin(

)2/sin()2/sin(

cDcDL

bDbDN

aDaDMnn GG

222*

)sin()sin(

)sin()sin(

)sin()sin(

LNMnn GG

2;

2;

2cDbDaD

where

GkD

2

You should already familiar with the functionof 2

)sin()sin(

M

Chapter 4

222*

)sin()sin(

)sin()sin(

)sin()sin(

LNMnnI GG

Consider

1st min occurs at0)sin(;0)sin(;0)sin( LNM

LNM ;;i.e.

GkDcDbDaD 2;

2;

2;

2

MaGk 2)2(

LcGk 2)2(

NbGk 2)2(

MaGk 1)

2(

LcGk 1)

2(

NbGk 1)

2(

or

Therefore, for a finite crystal, the diffractedintensity is finite based on the condition below.

I 0 if

MaGk 1)

2(

LcGk 1)

2(

NbGk 1)

2(

Example: for a very thin sample

Ewald sphere construction

2;

2

'' kSkS

When G

diffraction occurs dueto “shape effect”.

𝑆 ′ −𝑆