Stanford Synchrotron Radiation Laboratory

X-ray DiffractionMike Toney

Stanford Synchrotron Radiation Laboratory

1. Crystals and crystal geometry (planes)2. Bragg’s law3. Reciprocal lattice & reciprocal space

• definition and examples• relation to diffraction

4. Diffraction intensities and crystallography5. Some examples from current research

• Macromolecular Crystallography• Powder Diffraction – Zeolites• Strain: Eu films• Pentacene films

Outline

• B Warren, “X-ray Diffraction”, Dover (1990): $11.87 & Eligible for FREE Super Saver Shipping on amazon.com.

• BD Cullity & SR Stock, “X-ray Diffraction”, Prentice Hall (2001).

• J Als-Nielsen & D McMorrow, “Elements of Modern X-ray Physics”, Wiley (2001).

Bibliography

Diffraction vs Scattering

0 20 40 60 80

1000

2000

3000

4000

5000

6000

7000

21 24 27

2000

4000

6000

Inte

nsity

2θInte

nsity

2θ

diffraction: Bragg peaks

scattering: the rest

Crystal Lattice

Crystal lattice: periodic, repeating arrayLattice point – atom, molecule, or fixed arrangement of atoms, or protein, fullerene, …

Lattice Vectors span the unit cellunit cell

Crystal Lattice

a0

Crystal Planes

plane spacings (d):

(100): d = a0

(110): d = a0/√2

(111): d = a0/√3

(hkl): d= a0/√h2 + k2 + l2

a0

Bragg’s Law

n λ = 2d sin (θ)

λ = incident wavelength (=hc/E)

d

θ

Bragg’s Law: constructive interference of reflected X-rays

θ

2θ = scattering angle

2θ

Bragg’s LawAu: a0 = 4.0786 Å

d(111) = 2.3548 Å

for λ = 1.5498 Å, 2θ = 38.425 deg.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

37.4 37.6 37.8 38 38.2 38.4 38.6 38.8 39 39.2

2 Theta

Inte

nsity

(111)

Bragg’s Law

need:

• crystal oriented to give diffracting planes perpendicular to incoming & diffracted X-rays

• incoming X-rays at θ

• diffracted X-rays at θ

• AND n λ = 2d sin θ

n λ = 2d sin (θ)

Reciprocal Lattice

Reciprocal: “complementary”Webster’s Dictionary

Reciprocal lattice (space): language to describe diffraction and scattering

R K

The reciprocal lattice is set of vectors K so that:

e i K R = 1,

where R is any lattice vector

Reciprocal Lattice

a0

a* = 2π/a0

b* = 2π/a0

(-20) = (2 0)

e i K R = 1,

where R is any lattice vector

(20)(00) (10)

(01)

(-20)

(02)

a*

b*

Reciprocal Lattice

(20)(00) (10)

(0-1)

(0-2)

G1 or a*

G2 or b*

G1 d1 = 2π

G2 d2 = 2π

Reciprocal Lattice

(110)

(000) (100)

(111)

a0

real space:simple cubic

2π/a0

reciprocal space:simple cubic

e i K R = 1,

where R is any lattice vector

Reciprocal Lattice & Diffraction

Q

sample

kin kout

Q = scattering vector= kin – kout

• Constructive interference of X-rays (diffraction peaks) requires that e i Q R = 1.

• So Q = K, where K is a reciprocal lattice vector

(2 0)(-1 1)

(-1-1)

(1 1)

(1 -1)

(-2 0)

Q

Reciprocal Lattice

Stringent condition!

• Rotating crystal (macromolecular crystallography)

• Powder diffraction

• Live with it (single crystal)

(2 0)(11), (1-1)

(2 0)(-1 1)

(-1-1)

(1 1)

(1 -1)

(-2 0)

Q

Diffraction: Q = K

Diffraction Intensities

I have neglected something important.

What have I missed?

Diffraction intensities: These depend on atomic positions within the unit cell. => Crystallography

ClNa

2θ (deg)

inte

nsity

Diffraction Intensities

This can be reduced to:

where

and h,k,l are peak indices.

F = fn e 2 π i( ) hx n + ky n + lz n( )

n∑

r r n = xn + yn + zn( )

The structure factor, F, is defined as:

where fn is the atomic scattering factor;

is the position of the nth atom in the unit cell. r r n

nrQi

nnefF

rr⋅∑=

Intensity = |F|2

Phase problem

Explain better & show x, y, z and h k l

Diffraction IntensitiesF = fn e 2 π i( ) hx n + ky n + lz n( )

n∑

Simple cubic lattice:(x, y, z) = (0, 0, 0) => F = fn

bcc lattice:(x, y, z) = (0, 0, 0) & (½, ½, ½)⇒ F = fn {1 + eiπ(h+k+l)}⇒ F = 2 fn for h+k+l = even

= 0 otherwise

Diffraction Intensities

fcc lattice:(x, y, z) = (0, 0, 0), (½, ½, 0), (½, 0, ½) & (0, ½, ½)⇒ F = fn {1 + eiπ(h+k) + eiπ(h+l) + eiπ(k+l)}⇒ F = 4fn: h,k,l all even or odd

= 0, otherwise

Polyatomic Crystal: Rock Salt

ClNa

like fcc lattice:Na at (0, 0, 0)Cl at (½, 0, 0)⇒ F = {fNa + eiπh fCl}Ffcc

⇒ F = 4{fNa+ fCl}Ffcc: h,k,l even= 4{fNa- fCl}Ffcc: h,k,l odd= 0, otherwise

Macromolecular Crystallography

Macromolecule = big molecule, mostly biological.

Protein, virus, vitamins, …

Molecular structure determines function, how it works

astaxanthinpigment

= oxygen= carbon= hydrogen

Macromolecular Crystallography

angle

X-rays from SSRL

Vitamin B12

Dorothy Hodgkin (1910-1994)Nobel Prize, 1964

Vitamin B12Solved in 1956

Powder Diffraction

“Powder” average by random orientation of crystals

Problems due to peak overlap

Powder Diffraction

SSRL 2-1:

Dedicated for powder diffraction

incident x-rays

diffracted x-rays

sample

Powder Diffraction: Zeolites

Zeolites: microporousaluminosilicates used as catalysis, adsorbents, ….

Rare single crystals

A Burton et al., JACS 125, 1633 (2003)

SSZ-58: novel zeolitedeveloped by Chevron-Texaco Research

SSZ-58 structure ‘solved’with advanced ab initiomethods

Powder Diffraction

A Burton et al., JACS 125, 1633 (2003)

10-membered ring pores

huge surface area

Diffraction Measurements of Strain

Many materials properties depend on strain: magnetostriction, mobility, pizeoelectricity

X-ray diffraction provides a very accurate & precise method of strain measurement

Q = (4π/λ) sin θtypical 2θ resolution is 0.001-0.1 deggives Q resolution of 0.0001-0.01 Å-1

translates to d resolution of 0.005-0.00005 Å(0.5-0.005 pm)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

37.4 37.6 37.8 38 38.2 38.4 38.6 38.8 39 39.2

2 Theta

Inte

nsity

Symmetric

Asymmetric

∆ 2θ = 0.03 deg

Eu Thin Films: Strain

S. Soriano, K. Dumesnil, C. Dufour, T. Gourieux, A.Stunault, M. Hennion, J.A. Borchers, Ph.Mangin, Laboratoire de Physique des Matériaux (LPM) – Nancy, Intitut Laue Langevin (ILL) – ESRF - Grenoble,Laboratoire Leon Brillouin (LLB) (Saclay), NIST Center for Neutron Research (NCNR)

15nm (110)Nb 150°C

Y

(1120) Al2O3

50nm (110)Nb 800°C

(110)Eu 150°C

Molecular beam epitaxy (4.10-11 Torr)

Bulk Eu: bcc

TN = 90K

Para.

[001]

[100]

[010](110)

Eu(110)

(100) (010)

(001)

• Disappearance of one of the magnetic helices (τ//[001]) in favor of the two others: τ//[100] and τ//[010]

• Thermal hysteresis

375 nm thick filmsM

agne

tic p

opul

atio

ns

Temperature (K)0 10 20 30 40 50 60 70 80 90

0,0

0,1

0,2

0,3

0,4

0,5

0,6

(100)(010)(001)

RXMSESRF - BM28

Eu L2 edge

Magnetic State of Eu Thin Films

(110)

(100)(010)

(001)

• a⊥ = a(110): match bulk 300K-10K

• a//= a(002) =a(110):- match bulk T>Tcl- for T<Tcl constant to 10 K

• Tcl depends on the film thickness

• The amplitude of the strains increase with decreasing temperature and reduced film thickness.

0 50 100 150 200 250 300

4,54

4,55

4,56

4,57

4,58

4,59

750 nm

375 nm

75 nm

Temperature (K)

latt

ice

para

met

er (a

ngst

rom

)Eu Thin Films: Lattice Clamping

a (110)

a(001) & a(110)

neutron diffraction data

Eu Thin Films: Strain

Karine Dumesnil, UniversitéH. Poincaré - Nancy, France on SSRL beam line 7-2

15nm (110)Nb 150°C

Y

(1120) Al2O3

50nm (110)Nb 800°C

(110)Eu 150°C

• Use diffraction to accurately assess strain state

• Depth dependent strain

Eu Thin Films: Strain

8.22 8.23 8.24 8.25 8.26 8.27 8.28 8.290

5

10

15

20

25

30

35

40

87.5K

70K

88K

88.5K

Inte

nsity

(cts

/mon

)

(116)

8.22 8.23 8.24 8.25 8.26 8.27 8.28 8.290

5

10

15

20

25

30

35

40

45

50

(116)Decrease in T

Inte

nsity

(cts

/mon

)

89K 87K 85K 83K 81K 79K 76K 73K 70K

Q along (001) ( Å-1)

decreasing temperature

(110)(116)

(001)

Q along (001) ( Å-1)

Eu=> two domains with different strain states

45 50 55 60 65 70 75 80 85 90 95 100105110115120

4.556

4.558

4.560

4.562

4.564

4.566

4.568Omega = 0 / C component

along [200] along [002]

latti

ce c

onst

ant (

angs

trom

)

Temperature (K)

45 50 55 60 65 70 75 80 85 90 95 100105110115120

4.556

4.558

4.560

4.562

4.564

4.566

4.568Omega = 0 / A component

along [200] along [002]

latti

ce c

onst

ant (

angs

trom

)

Temperature (K)

Eu Thin Films: Strain

Domain A: (001) compressed compared to (100) and (010)

(110)

(001)

(100) (010)

Domain C: (100) and (010) compressed compared to (001)

Organic Thin Films

Organic FETSandra Fritz,C. Daniel Frisbie,Mike Ward,Chemical Engineering and Materials ScienceUniversity of Minnesota

Pentacene Thin Films

2 7 12 17 222theta (degrees)

Inte

nsity

(arb

itrar

y un

its)

“thin film” phase, d001 = 15.4Å-rocking curve width >0.05o

=> highly textured films

“bulk” phase, d001 = 14.4Å

• Thermal evaporation of pentacene

• Film morphology and structure dependant on substrate temperature, deposition rate and film thickness

1.5 nm - one layer

30 nm – ca 15 layers

5 µm

5 µm

300 nm SiO2

Si

1.5 – 60 nm pentacene

θ/2θ XRD: ca 50 nm film

Grazing Incidence Diffraction

2θα

β

2θ is scattering angleQ = scattering vectorQ = k’ - kQ = (4π/λ) sin θα = incidence angleβ = exit angle

SiO2

pentacene

Vary penetration depth by changing incidence angle α

Needs good collimation

300 nm SiO2

Si

pentacene

Pentacene Thin Films

1.2 1.7 2.2 2.7 3.2 3.7qxy (Å-1)

Inte

nsity

(a.u

.)

monolayer200 Å

(11L)

(02L)(12L)

(20L)

(21L)

(24L)(14L)

(23L)(13L)

2θα

β

b*

c*

a*

(-2 0)

Qz

Qxy

(-2 0 0)

(-2 0 1)

(-2 0 2)

} c*

Qxy

(2 0)(-1 1)

(-1-1)

(1 1)

(1 -1)

Fritz et al., unpublished

Pentacene Thin Films

} c*Qz

Qxy

(2 0 0)

(2 0 1)

(2 0 2)

b*

c*

a*

(-2 0 1)

(-2 0 2)

(-2 0 3)

2a*-2a*

02Lscan

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

qz (Å-1)

Inte

nsity

(a.u

.)

c * = .408 Å-1

(0 -2 1)

(0 2 0)

(0 -2 2) (0 2 1)

} c*Qz

Qxy

(2 0 0)

(2 0 1)

(2 0 2)

(-2 0 1)

(-2 0 2)

(-2 0 3)

2a*

Pentacene Thin Films

b*

c*

a*

11Lscan

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

qz (Å-1)

Inte

nsity

(a.u

.)

(1 1 0)

(-1 1 0)

(1 -1 1)

(-1 1 1) (1 1 1) (1 -1 2)

Qz

Qxy

(1-1 0)

(1-1 1)

(1-1 2)

(-11 1)

(-11 2)

(-11 3)

-a* + b*a* - b*

} c*Qz

Qxy

(1 1 0)

(1 1 1)

(1 1 2)

(-1-11)

(-1-12)

(-1-13)

-a* - b* a* + b*

|a* +/- b* |

Qz

Qxy

(11 0)

(-11 1)(1-1 1)

(-11 0)

02Lscan

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

qz (Å-1)

Inte

nsity

(a.u

.)

c * = .408 Å-1

(0 -2 1)

(0 2 0)

(0 -2 2) (0 2 1)

Pentacene Thin Films

1.2 1.7 2.2 2.7 3.2 3.7qxy (Å-1)

Inte

nsity

(a.u

.)

monolayer200 Å

(11L)

(02L)(12L)

(20L)

(21L)

(24L)(14L)

(23L)(13L)

2θα

β

lattice parameters:a = 5.933 (3) Åb = 7.540 (3) Åc = 15.51 (1) Åα = 93.2 degβ = 95.9 degγ = 90 deg

b

c

a

11Lscan

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

qz (Å-1)

Inte

nsity

(a.u

.)

(1 1 0)

(-1 1 0)

(1 -1 1)

(-1 1 1) (1 1 1) (1 -1 2)

Fritz et al., unpublished

β α

γ

• Pentacene thin films on SiO2 are crystalline, but distinct from bulk

• Thin Film: herringbone motif with molecules tilted a few-ten degrees

5µm

a

bγ

Fritz et al., unpublished

Pentacene Thin Films

Summary

Crystals and geometryBragg’s lawReciprocal lattice & reciprocal space• definition and examples• relation to diffractionDiffraction intensities and crystallographyFew Examples• Molecular & Crystal structure• Strain