Diffraction from Crystals “Shape Factor”phome.postech.ac.kr/user/aemn/download/lec_04.pdf · 2014-07-24 · Lecture 3 Diffraction from Crystals “Shape Factor” - Fultz & Howe,

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Lecture 3

Diffraction from CrystalsDiffraction from Crystals

Lecture 3

Diffraction from Crystals“Shape Factor”

Diffraction from Crystals“Shape Factor”

- Fultz & Howe, Chap. 5- Williams & Carter, Chap. 17- Reimer, Chap. 7, p

AMSE 609 Spring 2010

Shape Factor

exp 2lattice

k F k i k r• Scattered wave:

“Crystal size effect”

exp 2lattice

gS k i k r

exp 2gr

gk F k i k r• Scattered wave:

• Shape factor: pgr gp

z

Nzc

Unit cell(/w a, b, c

lattice - Location of each unit cell:

yz

NxaCrystal

(N N N unit cells)

lattice parameters)

Location of each unit cell:

x y zn n n gr ax by cz

- Δk components:

x Nyb

(Nx Ny Nz unit cells)

x y z k k x k y k z

11 1NN N

11 1

0 0 0

11 1

exp 2yx z

x y z

yx z

NN N

x x y y z zn n n

NN N

n n n

S k i k a k b k c


0 0 0

exp 2 exp 2 exp 2yx z

x y z

x x y y z zn n n

n n n

i k a i k b i k c

Shape Factor

11 1

0 0 0

exp 2 exp 2 exp 2yx z

x y z

NN N

x x y y z zn n n

n n n

S k i k a i k b i k cx y z

• Each sum is a truncated geometric series of the form:

2 3 4 11 ... Nr r r r r S

1 ...

1

1

N

r r r r r

r

r

S

exp 2 xr i k awhere, e.g.,

1

0

1 exp 2exp 2

1 exp 2

xx

x

Nn x x

xn x

N

i k a

i k ai k a

* 1 exp 2 1 exp 2

1 exp 2 1 exp 2x x x x

x x

N NS S

x

i k a i k ak

i k a i k a

2*

2

sin

sinxN

S S

xx

k ak

k a


sin xk a

Shape Factor• The denominator varies slowly with respect to the numerator,

approximation leads to:

2 2*

22

sin sin

sinx xN N

S S

x x

xx x

k a k ak

k a k a

pp

2

1E

x

x

kk a

• The envelop function: xk a

• As N becomes large:As N becomes large:

- Height of main peak N2

- Width of main peak (aN)-1

H i ht f t llit k - Height of satellite peaks

- Satellite peaks get closer


Shape FactorpThin foil effect – ‘Relrod’

• Reciprocal Lattice Rods (Relrod)Reciprocal Lattice Rods (Relrod)

- Streaks may also occur along di ti if th x or y direction if the

dimension is small.

- Streaks are the reason why exact orientation of a crystal exact orientation of a crystal is not possible with spot diffraction pattern.


Shape Factorp

2

2

22

2

sin

ix

I

NF

x

k k

k ak

k

2

2 2

22

sin

sin sin

sinsiny y z z

N N

xk a

k b k c

k ck b sinsin zy k ck b


Shape Factorp• Example: Guinier-Preston (GP) zones - Al-4%Cu alloy

- Fe-2.9%Mo alloyy

2g

disk

Al Cu Alr

whole

F F gk k k i k r


2g

whole

Alr

F gk i k r

Deviation Vector• Express Δk as the difference of an exact reciprocal lattice

vector, g, and a “deviation vector”, s:

k g s

g k s x y z s s x s y s z*

2 2lattice lattice

S k i k i ( i t )

• Effect on Shape Factor:

exp 2 exp 2g gr r

g gS k i k r i g s r

exp 2 integer exp 2 exp 2lattice lattice

S k i is r is r

( integer) gg r

exp 2 integer exp 2 exp 2g gr r

g gS k i is r is r

S S k s

• Effect on Structure Factor:

F Fk g ( is small)ks r


F Fk g ( is small)ks r* k g s , more general

Kinematical Intensityy

2

22 sin

I

s N

k k

a

2

2

2 2

sin

sin

sin sin

xs NF

s

s N s N

x

x

ag

a

b c

22

sin sin

sinsiny y z z

zy

s N s Nss

b c

cb

• Shape factor, S(s), depends only on s.

• Structure factor, F(g), depends only on g.


Deviation from Exact Bragg Conditiongg

k

s

0kg

• Exact Bragg condition:• Deviation, s:

gkk 0 Ssgkk 0

lattice latticeSince grg is an integer exp 2

g

lattice

r

gk F k i g s r exp 2g

lattice

r

gs F g is r

• Definition: - if s < 0: reciprocal lattice point outside Ewald sphere- if s > 0: reciprocal lattice point inside Ewald sphere


if s > 0: reciprocal lattice point inside Ewald sphere

Kikuchi Lines

Diffraction cone

sample

Diffraction cone

dhkl

αβ

dhkl

2θ=β - α

dark brightscreen


Kikuchi LinesPair of a “bright” and “dark” line

21121

)1(

)1(1

IIcIcIcII

IIcIcIcII K


21212 )1(2

IIcIcIcII K

Kikuchi Lines

• The dashed line right between the excess line and the deficient line marks the intersection of the reflecting planes with the plane of the diffraction patterng p p p

• The spacing Dhkl of the two lines corresponds to 2θB, thus Dhkl = 2LθB = L/dhkl

Dhkl equals the spacing between the spot of the transmitted beam and the spot of the (hkl) Bragg reflection

• when the specimen is oriented exactly for Bragg reflection at the (hkl) planes, the deficient Kikuchi line intersects the transmitted beam, while the excess Kikuchi line


,intersects the Bragg reflection

Kikuchi Lines

• Pairs of parallel lines consisting of one bright and one dark lines in diffraction mode

• Electrons are scattered elastically (diffraction spots) or inelastically (diffuse in all directions, with maximum intensity along incident beam direction and decreasing i t it ith i i l )intensity with increasing angle)

• The intensity decreases with increasing scattering angle.

• Inelastically scattered electrons (assumption: negligible energy-loss) act as a new primary beam which can undergo Bragg diffraction (in all three dimensions) causing diffrcation conesdiffrcation cones

• Kikuchi lines exist only in thick samples

• Can be used for accurate determination of orientation (ca. 0.1degree instead of typically 4 degrees in spot patterns)


Diffraction Condition

position of lattice plane

• Exact Bragg (strong “two-beam”condition)condition)

s =0 (dynamic theory required!)

Diffraction pattern

s<0• Symmetric case

(Laue condition)

s < 0 with s = θgs < 0 with s = -θg(kinematic theory !)


position of lattice plane

Kinematical vs. Dynamical

I << I (weak scattering)

• Kinematical theory

y exp 2

g

lattice

r

gk F k i k r

- Id << I0 (weak scattering)

- Thin samples

- Small deviation from exact Bragg condition (required):

tt d t S t diff b it ti f i l l tti scattered vector S must differ by an excitation error s from any reciprocal lattice g

Δk = g + s

- Amplitude A of the diffracted beams are summed up taking phase shifts into account

A2

A3

exp 2lattice

gs F g is r

A1• Dynamical theory

- Id ≈ I0 (multiple scattering)

gr g

- Thick samples

- Exact Bragg condition

- Considers diffraction from beam diffracted back into the primary beam


- Uses Schrödinger equation (Howie-Whelan Eqn)

Home Work• Fultz & Howe book (3rd ed.): Problems 5.1, 5.4, 5.10, 5.16, 5.18.• Due date: April 6th.Due date: April 6 .


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Diffraction from Crystals “Shape Factor”phome.postech.ac.kr/user/aemn/download/lec_04.pdf · 2014-07-24 · Lecture 3 Diffraction from Crystals “Shape Factor” - Fultz & Howe,