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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
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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
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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
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Representation of TextureStereographic Projections
Z
North Pole
South Pole
ReferenceSphere
5
f
y
y/2
Z
X
Y
[001] || Z
P
P‘
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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
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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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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
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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)
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g-TiAl alloy
Bob He, Bruker (2011)
Determination of Texture from 2D Measurements
(111) Pole
Figures
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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
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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
19
Typical for epitaxial thin films {100} <100>TexturX
Y
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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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TYPES OF TEXTURES
Deformation Textures in Mecanically-cycled
NiTi Shape Memory Alloy
211Zotov (2014)
200
Individual Pole Figures
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Types of Textures
Cold-rolled textures
Typical fcc Texture Components
(111) (200)
(111) (200)
Leffers & Ray (2009)
Leffers & Ray (2009)
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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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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
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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
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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
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Examples of Grazing Incidence Diffraction
Ti coated with Hydroxyapatite (HA)
Large a
Small a
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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)
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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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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
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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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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
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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
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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
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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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