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Introduction to Diffraction analysis
Luca Lutterotti
Department of Materials Engineering and Industrial Technologies
University of Trento – Italy
Outline: basic concepts
• The Bragg law
• The intensity of the diffraction
• Powder diffraction and instrumentation
– Bragg-Brentano
– Texture goniometers
– Residual stress measurements
• Diffraction analyses
The Bragg law
• Constructive interference and interplanar spacing:
Reciprocal space
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The Bragg law and the intensities
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• Consider diffraction from the (001) plane
Diffraction intensities
• The intensity in a powder diffractometer
• The structure factor:
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Atomic scattering factor and Debye-Waller
• The atomic scattering factor for X-ray decreases with the
diffraction angle and is proportional to the number of electrons. For neutron is not correlated to the atomic number.
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Neutron scattering factors
• For light atoms neutron scattering has some advantages
• For atoms very close in the periodic table, neutron scattering may help distinguish them.
Neutron & X-Ray Scattering Lengths
Q, Å-1
0
5
10
15
20
25
30
0 2 4 6 8
f,e-
CoFe
Ti
Co-6e-
MgO
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-0.40
0.00
0.40
0.80
1.20
Element
bx1
0
cm
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H
Be
Sc
Ti Mn
Xe
NiFe
CoV
Li
N Cl
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X-ray and neutron diffraction
10.0 0.05 155.9 CPD RRRR PbSO4 1.909A neutron data 8.8
Scan no. = 1 Lambda = 1.9090 Observed Profile
D-spacing, A
Cou
nts
1.0 2.0 3.0 4.0
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1
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1
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2
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2
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Scan no. = 1 Lambda1,lambda2 = 1.540 Observed Profile
D-spacing, A
Cou
nts
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E 4
.0
.
5
1.0
1.5
X-ray Diffraction - CuKaPhillips PW1710• Higher resolution• Intensity falloff at small dspacings• Better at resolving small lattice distortions
Neutron Diffraction - D1a, ILL !!!!=1.909 Å• Lower resolution• Much higher intensity at small d-spacings• Better atomic positions/thermal parameters
Compare x-ray & neutron powder patterns (both CW)
Thermal or Debye-Waller factor
• It causes a decrease of the
intensities at high angle
• It is proportional to the thermal
vibrations
• Intensities decrease increasing
the temperature
• From the Debye-Waller it is
possible to estimate the Debye temperature
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Powder diffraction and Debye-Scherrer camera
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N I*(%&'()'13(*^#1$'(/0*-612')'3%(213(5'(61$@'(+,$(*,)'()1%'$016*(D1A'$1@'(/0*-612')'3%(_8C`(a(+,$(I6(1%($,,)(%')-F(%&'(%')-'$1%#$'(+12%,$(213(6'1/(%,(1(/$1)1%02(/'2$'1*'(03(/0++$12%'/(03%'3*0%7H(-1$%02#61$67(.&'3(%&'(*1)-6'(0*(1%(&0@&(%')-'$1%#$'
N <,**(,+(03%'3*0%7(0*(),*%(,5A0,#*(1%(&0@&(13@6'*
N I*(-'1K*(1$'(*#--$'**'/(57(%&'$)16(),%0,3(%&'(512K@$,#3/($0*'*(/#'(%,(%&'$)16(/0++#*'(*21%%'$03@(
A modern diffractomer
Page 7.7 © 1999 Scintag Inc. All Rights Reserved.
Chapter 7: Basics of X-ray Diffraction
GONIOMETER
The mechanical assembly that makes up the sample holder, detector arm and associated gearing is referred to as go-
niometer. The working principle of a Bragg-Brentano parafocusing (if the sample was curved on the focusing circle
we would have a focusing system) reflection goniometer is shown below.
The distance from the X-ray focal spot to the sample is the same as from the sample to the detector. If we drive the
sample holder and the detector in a 1:2 relationship, the reflected (diffracted) beam will stay focused on the circle of
constant radius. The detector moves on this circle.
For the THETA : 2-THETA goniometer, the X-ray tube is stationary, the sample moves by the angle THETA and the
detector simultaneously moves by the angle 2-THETA. At high values of THETA small or loosely packed samples
may have a tendency to fall off the sample holder.
A typical spectrum
Parafocusing circle (Bragg-Brentano)
Bragg-Brentano Diffractometer – “parafocusing”
Diffractometer
circle
Sample
Receiving slit
X-ray sourceFocusing circle
Divergent beam optics
Incident beam
slit
Beam footprint
Theta-theta diffractometer
Page 7.8 © 1999 Scintag Inc. All Rights Reserved.
Chapter 7: Basics of X-ray Diffraction
GONIOMETER
For the THETA:THETA goniometer, the sample is stationary in the horizontal position, the X-ray tube and the de-
tector both move simultaneously over the angular range THETA.
Slits system in Bragg-Brentano
soller slits
sample
divergenge slit
soller slits
receiving slit
antiscatter slitscatter slit
tub
e ta
rget
dete
cto
r
Texture goniometer
Page 7.14 © 1999 Scintag Inc. All Rights Reserved.
Chapter 7: Basics of X-ray Diffraction
APPLICATIONS
coordinate system from a position coincident with the specimen coordinate system to a
given position.
The ODF is a function of three independent angular variables and gives the
probability of finding the corresponding unit cell (lattice) orientation.
Polefigure data collection : The systematic change in angular orientation of the sample is normally achieved by
utilizing a four-circle diffractometer. We collect the intensity data for various settings
of CHI and Phi. Normally we measure all PHI values for a given setting of CHI, we
then change CHI and repeat the process.
Four circle diffractometer
Texture orientations
Page 7.20 © 1999 Scintag Inc. All Rights Reserved.
Chapter 7: Basics of X-ray Diffraction
TEXTURE ANALYSIS
The determination of the preferred orientation of the crystallites in a polycrystalline aggregate is referred to as tex-
ture analysis. The term texture is used as a broad synonym for preferred crystallographic orientation in a
polycrystalline material, normally a single phase. The preferred orientation is usually described in terms of pole fig-
ures.
The Pole Figure : Let us consider the plane (h, k, l) in a given crystallite in a sample. The direction of the plane
normal is projected onto the sphere around the crystallite. The point where the plane normal
intersects the sphere is defined by two angles, a pole distance a and an azimuth ß. The azimuth
angle is measured counter clock wise from the point X.
Let us now assume that we project the plane normals for the plane (h, k, l) from all the crystallites irradiated in the
sample onto the sphere.
Each plane normal intercepting the sphere represents a point on the sphere. These points in return represent the
Poles for the planes (h, k, l) in the crystallites. The number of points per unit area of the sphere represents the pole
density.
X
N
a = 55°
ß = 210° D
( h, k, l)
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Pole figure representation
Pole figure projections (and inverse)
Residual stress measurement
Goniometer
PSD detector
X-ray tube and
collimator
Psi movement
Phi movement
X-ray generator
Electronic for PSD
Ratemeter
Software
Residual stress analysis
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Diffraction analyses
• Phase identifications (crystalline and amorphous)
• Crystal structure determination
• Crystal structure refinements (cell parameters and atomic
positions)
• Quantitative phase analysis (and crystallinity determination)
• Microstructural analyses (crystallite sizes – microstrain distributions etc.)
• Texture analysis
• Residual stress analysis
• Order-disorder transitions and compositional analyses
• Thin films
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