diffraction methods in material science · outline of today‘s lecture texture analysis pole...

DIFFRACTION METHODS IN MATERIAL SCIENCE

PD Dr. Nikolay Zotov

Email: zotov@imw.uni-stuttgart.de

Lecture 7

OUTLINE OF THE COURSE0. Introduction

1. Classification of Materials

2. Defects in Solids

3. Basics of X-ray and neutron scattering

4. Diffraction studies of Polycrystalline Materials

5. Microstructural Analysis by Diffraction

6. Diffraction studies of Thin Films

7. Diffraction studies of Nanomaterials

8. Diffraction studies of Amorphous and Composite Materials

2

OUTLINE OF TODAY‘S LECTURE

Texture Analysis

Pole Figures

Measurement of Pole Figures

Characteristics of Textures

Examples

Diffraction Studies of Thin Films

Grazing Incidence X-ray Diffraction (GIXRD)

X-ray/Neutron Reflectivity

3

TEXTURE ANALYSIS

Texture is the distribution of the orientations of

grains in a polycrystalline sample

Orientation Distribution Function (ODF)

ODF(g) = 1/V ∂V(g)/∂g; g =f, Q,y - Euler angles describing the orientaion of the

sample

Every colour – different crystallite orientation

4

Relative volume fraction of crystallites

with orientation g, g + dg

Representation of TextureStereographic Projections

Z

North Pole

South Pole

ReferenceSphere

5

f

y

y/2

Z

X

Y

[001] || Z

P

P‘

6

Representation of Texture – Pole Figures

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Representation of TexturePole Figures

Measuring grid

Bunge

Pole figure = variation in the diffracted intensity as a

function of the orientation of the

crystallites given as points on a stereographic projection.

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Bragg equation: 2dhklsin(Q) = l

1 22

Q Q

AB

Tilt of Sample y

Source Detector

3

(hkl)

Measurements of Texture

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measured Intensities: Ihkl(f,y)

Intensity of powder specimens

Ihkl ~ V (in general)

Ihkl(f,y) ~ V(f,y)

The intensity at every point (f,y) is proportional to the Volume of the crystallites

with this orientation

Reflection geometry

Measurements of Texture

Measurements of Texture

X-rays, neutrons (Monochromatic Beam)

# Eulerian Cradle

# Point Detector or

2D detector (Image Plate, CCD)

Neutrons Time-of-Flight

X-ray Surface texture

Electrons

Neutrons Bulk Texture

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Texture Measurements with Eulerian Cradle

and Point Detector

Modes w - f

y(c) - f

Reflection geometry (Vertical Scattering Plane)

Full Eulerian

Cradle

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12

Texture Measurements with Eulerian Cradle

and Point Detector

X-ray Tube

Colimator

for parallel

beam

1/4 Eulerian Cradle

X-Y-Z Table

Scinti

Detector

Graphite

MonocromatorColimator

f

Texture Measurements with Image Plate (CCD)

Reconstruction of standard pole figures from Intensities along the Debye rings

measured at different w; Mapping (h,w) → (f,y)

13

g-TiAl alloy

Bob He, Bruker (2011)

Determination of Texture from 2D Measurements

(111) Pole

Figures

14

Texture Measurements with Neutrons (TOF)

● Only w rotations necessary

● Simultaneous measurement of different scattering angles at different banks (panels) of detectors

→ simultaneous measurement of different pole figures

● Measurement of Bulk Texture

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w

Limestone, LANSCE (USA)

Wenk (2001)

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Limestone

Reconstruction of Pole Figures from Neutron Diffraction Experiments

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Characteristics of Textures

● Types

‚Random‘ Texture (no prefered orientation)

Fiber Texture

‚Single-Crystal-like‘ Texture

Deformation Textures in cold(hot)-rolled metals/alloys

(Distribution of grains with a given hkl)

● Strength of Texture

(Number of grains with a given orientation)

● Shrapness of Texture

(Variations of the individual grains around the average orientation)

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Types of Texture

‚Single-Crystal-like‘ Textures

100

0-10 010

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Typical for epitaxial thin films {100} <100>TexturX

Y

Ag 200

Fiber Texture

(crystallites tilted ~ 55o with

respect to the surface with random

orientation in the plane of film)

‚Single-Crystal‘ Texture

(crystallites oriented mostly with (100) planes

paralell to the surface)

Types of Texture

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110

TYPES OF TEXTURES

Deformation Textures in Mecanically-cycled

NiTi Shape Memory Alloy

211Zotov (2014)

200

Individual Pole Figures

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22

Types of Textures

Cold-rolled textures

Typical fcc Texture Components

(111) (200)

(111) (200)

Leffers & Ray (2009)

Leffers & Ray (2009)

Cold-Rolled Austenitic Steel

Morikawa et al., Mater. Trans. (2010)

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Examples of Strength and Sharpness

Stronger/sharper Weaker/more diffuse

Ag 200

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Bachmann et al. (2012)

Single-Crystal TextureNiW (111) Textures

Sharpness of Texture increases with annealing time

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26

f = f1

y = F

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Orientation Distribution Functions

Brass Deformation Texture

ODF(f1,F,f2)

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TYPES OF TEXTURES

Deformation Textures in Mecanically-cycled

NiTi Shape Memory Alloy (BCC)

ODF

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Calculation of ODF

requires at least 3

different pole figures

Classification according to Dimentionallity

Bulk Materials (single crystals)

Polycrystalline/Microcrystalline Materials

Thin Films (polycrystalline; ‚single-crystal‘ or amorphous)

Single-Layer

Multilayer

Nanostructures

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Specific Diffraction Methods for Thin Films

Small thickness of the TF → Small Diffraction Volume

Weak signal/Noise ratios

Strong Effect of the Substrate

● Grazing Incidence

● X-ray/Neutron Reflectivity

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Specific Diffraction Methods for Thin FilmsPenetration Depth

0 20 40 60 80 100 120 140

10-2

10-1

100

a=2Q/2

a=20o

a=10o

a=5o

a=2o

a=1o

Pe

netr

atio

n d

ep

th (

mm

)

Diffraction angle (o2Q)

Penetration Depth (63% absorption)

Reflection geometry; af ≠ ai

sin(ai)sin(2Q-ai)

t63 ~ -----------------------------

µ[sin(ai) + sin(2Q-ai)]

Gold, CuKa,

m 4000 cm-1

aiaf = 2Q -ai

Reflection geometry

af = ai = Q

I = IoA(Q) = Io[1-exp(-2µt/sin(Q)]

0.63 = I/Io = 1-exp(-2µt63/sin(Q)]

t63 ~ sin(Q)/2µ

Grazing Incidence Method

Principle

● Relatively large wavelength (small absorption)

● Stationary Primary Beam making

very small angle with the sample (0.1 – 5o)

● Only Detector (2Q) Scan

Conventional Geometry/Scan Q/2Q

2Q

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Parallel Beam

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Grazing Incidence Method

Principle

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Grazing Incidence Method

Principle

Examples of Grazing Incidence Diffraction

Ti coated with Hydroxyapatite (HA)

Large a

Small a

36

CdSSe on Graphite Substrates

Only Graphite Peaks!Q-2Q scan

Grazing Incidence

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Ti Anodization

Kosanovic (2012)

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In-situ Growth of Ag and Sn Thin Layers

Grazing Incidnce

16 18 20 22 24 26 28 30 32 34

0

1000

2000

3000

4000

5000

6000

Tim

e (

s)

2T (degrees)

15.00

160.6

321.3

481.9

642.5

803.1

963.8

1124

ANKA Synchrotron Source

l = 1.0 Å

a = 4o

Depostion first of Sn

Deposition of Ag on top

---------------------------------

Sn is textured I200 < I101

No Ag peaks!

Diret formation of Ag3Sn

Sn(200) Sn (101)

Ag3Sn(100) Ag3(020) Ag3Sn (012) Ag3Sn(221)

Sn(220) Sn(211)

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Grazing Incidence of aged In-Ag Bilayers

Ag

AgIn2

Ag2In

Rossi, Zotov (2016)

Applications of Grazing Incidence Diffraction

● Thin film Phase Analysis

● Oxidation products

● Corrosion Products

● Monitoring In-situ TF Deposition

● Near-Surface Depth Profiling

● Orientation of TF with respect to substrate

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42

X-ray/Neutron Reflectivity from TF and Multilayers

● ● ●

● ● ●

● ● ●

0

(hkl)

Q = 4psin(Q)/l

TF

ai

af = 2Q -ai

Q

Q << Ghkl

No diffraction!!!

Processes:

Reflection

Transmission

Absorption

Substrate

Vacuum/Air

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X-ray/Neutron Reflectivity from TF and Multilayers

n - Refractive index

d - Dispersion term

ß - Absorption term

d = (l2/2p) re r ; r Density of the material

ß = (l/4p) µ; re = 2.81 x 10-15 m

Transmited wave possible only if cos(at) ≤ 1; ai ≥ ac

Critical angle ac = (2d)½ ; ai ≤ ac Total external reflection

Z

Scattering vector: QZ = (2p/l)[sin(ai) + sin(af)]

Qc = (16prer)½ Iref= rr* = |r|2

r = Er/Eo

Snell Law cos(ai) = ncos(at)

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X-ray/Neutron Reflectivity from TF and Multilayers

Salamon et al. (2013)

Constructive interference of waves

reflected from the different layers (j)

Amplitude of total reflected wave

r = S rj,j+1 exp(iQZzj)

For large number of sharp layers

r ~ 4pre/QZ2 ∫ [∂r(z)∂z] exp (izQz) dz =

= 4pre/QZ2 FT [∂r(z)∂z]

R = |r|2 ~ 1/ QZ4 ~ 1/Q4

X-ray/Neutron Reflectivity from TF and Multilayers

r1

r2 Dr = │r1 - r2│

Q/2Q scans,

but both Q and 2Q are

small

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46

Effect of Surface/Interface Roughness

J. Daillant, A. Gibaud, X-ray and neutron reflectivity-

Principlesand Applications, p. 245

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Effect of Surface/Interface Roughness

Roughness – chemical gradients

geometrical roughness

Sardela (IUC)

Reflectivity Examples

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Kiessig fringes: Q2 – ac2 = m2(l/2D)2

Kiessig fringes

m is the number of the corrsponding maximum

m=1

m=2

Rafailovic et al. (2009)

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0,5 1,0 1,5 2,0 2,5 3,0 3,5 4,0 4,5 5,010

0

101

102

103

104

105

106

Inte

nsity (

a.u

.)

Diffraction angle (o2)

Si

Mo

Mo

Mo

r t [Å] s [Å]

0.68 19.6 5.8

0.93 236.5 34.0

1.09 14.1 2.7

1.00 5.0 2.7

1.00 2.8W

Edge of TER

Kiessig oscillations (fringes)

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Fiting of Reflectivity Data

X-ray Reflectivity Applications

Determination of Thicknesses

Determination of Interface Roughnesses

Density Fluctuations

Roughness Correlations

Determination of Refractive Indeces

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Sources

O. Engler, V. Randle, Introduction to texture analysis, 2000

H.J. Bunge, Texture analysis in material science, 1982

J. Daillant, A. Gibaud, X-ray and Neutron Reflectivity, Springer

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