Applications of X-Ray Diffraction
Dr. T. Ramlochan
March 2010
C o u n ts
1000
500
C 3 S - A li t e ( M 3 ) , N i s h i e t a l. 5 7 . 0 %
C 2 S - ß - B e li t e , M u m m e 1 8 . 3 %
C 3 A - A lu m i n a t e c u b i c 4 . 7 %
C 3 A - N a - A lu m i n a t e o r t h o 0 . 7 %
C 4 A F - B r o w n m i lle r i t e 1 0 . 0 %
M a g n e s i u m o x i d e - P e r i c la s e 2 . 0 %
C a lc i u m o x i d e - L i m e 0 . 1 %
P o t a s s i u m s u lf a t e , b e t a - A r c a n i t e 0 . 5 %
C a lc i t e 0 . 3 %
Q u a r tz 0 .0 %
G y p s u m 0 .0 %
C a l c i u m s u lf a t e h e m i h y d r a t e 0 . 5 %
A n h y d r i te 3 .3 %
P o r t la n d i t e 0 . 3 %
A p h t h i t a li t e 1 . 0 %
C a - la n g b e i n i t e 1 . 3 %
0
10
20
40
30
P o s i ti o n [° 2 T h e ta ]
50
60
Applications of powder diffraction
Diffraction pattern gives information about peak positions, intensity, and
shape
Intensit y

Position



Identification (qualitative) – most common use of powder diffraction is
identification of crystalline phases (search/match); peak positions and
intensity related to unique crystal structure
Quantification – determination of phase amounts in a polycrystalline
material; peak intensity and shape related to concentration
 RIR – reference intensity ratio
 Whole-pattern fitting (Rietveld analysis) – computationally intensive;
can only be applied with powerful analytical software.
Determination of crystallographic structure (i.e., unit cell) – much of Xray diffraction is concerned with discovering and describing crystal
structure
Phase identification

Diffraction patterns are unique fingerprints (distinct and reproducible) of
the crystal structure of materials that can be used to determine phase
composition of a polycrystalline material
 “…every crystalline substance gives a pattern; the same substance
always gives the same pattern; and in a mixture of substances
each produces its pattern independently of the others.”
-- A.W. Hull (1919), “A New Method of Chemical Analysis”
Gypsum (CaSO4·H2O) layered structure and diffraction pattern
Phase identification

Phase identification is essentially an exercise of pattern comparison
between the unknown and a database of single-phase reference
patterns
Diffractogram is reduced to a table of ‘d-I’ pairs (d-spacing and relative
intensity); some important information is lost

29.113[°]
11.634[°]
Counts
2000 0
20.725[°]
PDF #70-0982 Gypsum
47.869[°]
48.343[°]
48.826[°]
45.536[°]
46.262[°]
46.554[°]
46.997[°]
47.442[°]
43.650[°]43.424[°]
44.158[°]
44.579[°]
40.622[°]
42.170[°]
39.327[°]
37.431[°]
38.009[°]
36.055[°]
36.318[°]
36.603[°]
35.401[°]
34.541[°]
32.132[°]
32.734[°]
28.082[°]
25.056[°]
18.741[°]
16.505[°]
23.389[°]
33.366[°]
31.162[°]
1000 0
0
10
20
30
Position [°2Theta]
40
50
No.
1
3
4
5
7
8
9
10
12
13
15
17
18
21
22
23
24
25
27
29
32
33
34
36
37
38
h
0
1
-1
0
-1
-1
0
-2
0
1
2
-2
1
-1
-2
-1
-2
-2
1
2
0
-1
-3
-2
-1
-3
k
2
1
2
4
1
4
0
1
5
5
0
2
4
5
4
2
5
1
7
1
6
4
1
6
7
2
l
0
0
1
0
2
1
2
1
1
0
0
2
1
2
2
3
1
3
0
1
2
3
2
2
2
1
d[A]
7.601
4.731
4.282
3.800
3.175
3.065
2.868
2.783
2.683
2.595
2.489
2.453
2.401
2.219
2.141
2.082
2.072
2.049
1.990
1.949
1.899
1.881
1.864
1.812
1.805
1.795
2Theta[deg]
11.63
18.74
20.73
23.39
28.08
29.11
31.16
32.13
33.37
34.54
36.06
36.60
37.43
40.62
42.17
43.42
43.65
44.16
45.54
46.55
47.87
48.34
48.83
50.32
50.52
50.82
I[%]
100
1.5
96
11
3.2
53
38.5
6.5
25
4.1
8.6
5.1
3.7
8.7
1.2
14.5
7.9
3.2
2.1
1.7
10
9
2
9.1
5.1
4.1
Search/Match

Search/Match – legacy (manual) method used database of index cards
or books with d-I for reference materials
 Tedious, time intensive, human error
 OK for single phase, but multi-phase very difficult
(1) file number, (2) three strongest lines,
(3) lowest-angle line, (4) chemical
formula and name of the substance, (5)
data on diffraction method used, (6)
crystallographic data, (7) optical and
other data, (8) data on specimen, (9)
diffraction pattern.
Search/Match


International Center for Diffraction Data (ICDD) current Powder
Diffraction File (PDF-2 2006) database contains 186,107 entries of
almost every known inorganic (159,809) and many organic (28,610)
crystalline substances
 Includes cell parameters, d-spacing, chemical formula, relative
intensity, RIR
Software used to determine peak positions and intensities and used to
search/match or compare with all (or subset/restrictions) of the ICDD
PDF-2 database
 Some knowledge about the material can be used to limit search
 Reference patterns with high ‘score’ are visually compared with the
sample data and best match(es) are selected by user
 Works well for abundant phases, not so well for minor phases
 Human error
Phase identification problems


Accurate line position is very important
Specimen displacement
 geometry of diffraction requires that
specimen lie on the focusing circle and
be at the center of the diffractometer
circle or will cause angular errors (e.g.,
if sample is “high” the detected ∆2θ will
be positive)

Strain in crystal lattice
 Macrostrain causes uniform strain in
unit cell; unit cell dimensions and
distances between planes altered;
shifts the location of the diffraction peak
in the pattern
Quantitative phase analysis: Intensity ratios


Concentration (wt%) (and density) of a particular phase is proportional to its
intensity (peak area minus background, height?)
 More crystallites, more reflections, greater intensity
Klug and Alexander (1954) were first to describe a technique for
quantification using intensities of the crystalline phases in a mixture
 Ratio of peak intensity from unknown phase ‘A’ (I?) to a standard ‘B’ (IS)
is a linear function of the mass fraction of ‘A’ in the original sample
 Use known amount of internal standard mineral (e.g., rutile) to calibrate
the intensities of the unknown phase
Quartz
Rutile
26.646 °2Theta
27.434 °2Theta
Counts
10 00
500
0
26
27
Position [°2Theta]
28
Calibration (regression) curve
Quantitative phase analysis: Intensity ratios





Use single peak only, generally most intense peak (I100%)
Requires fully resolved peak
Chose internal standard – must not overlap any other peaks; simple
well-defined pattern (e.g., face centered cubic); same crystallite size as
sample
Spiking (standard addition) – add extra wt% of desired phase to mixture
and acquire at least 2 scans
External standard
Quartz
26.646 °2Theta
Counts
10 00
500
0
26
27
Position [°2Theta]
28
Calibration (regression) curve
Reference intensity ratio (RIR)

General formula for relating intensity ratio to mass fraction

ICDD PDF-2 uses corundum (Al2O3) as reference B and gives k for
50:50 mixture of phase A and corundum
 RIR is I/Icor using intensity of the strongest peak (100%)
 If I1/Icor is k1 and I2/Icor is k2, then I1/I2 is k1/k2
 If we know RIRs for every phase in mixture, we can determine the
relative amounts of each phase (do not need corundum) because
sum of all mass fractions equal 1
 Quick way to get ‘semi-quantitative’ information, but often
inaccurate due irregularities in sample
Problems with intensity ratios


Uses single peak (I100%)
Ideal sample is homogeneous and the crystallites have a random
distribution of all possible planes
 Each possible reflection from a given set of h, k, l planes will have an
equal number of crystallites contributing to it
 Can only occur if particles are spherical
 Crystal fragment shape is influenced by cleavage
 Platy or acicular crystals will have dominant direction when compacted
Cubic (110) (d2) family will have twice the intensity
of the (100) (d1) family due to multiplicity
Problems with intensity ratios

Non-random distribution of the crystallites is referred to as preferred
orientation (texture)
 Most common cause of deviation of experimental data from the ideal
intensity pattern
 Intensity ratios are greatly distorted by preferred orientation
 Minimise by back-loading sample, slurry with acetone
C o u n ts
N a tu r a l G y p s u m
G yp s u m P D F 7 0 -0 9 8 2
4000
2000
0
10
15
20
P o s i ti o n [° 2 T h e ta ]
25
30
35
Problems with intensity ratios
Compositional variations (e.g., site occupancy)
 Substitution of one atom for another in unit cell can alter intensities
(impure phases or solid solution)
12 00
C 4AF
C 4 A 1 .8 F 0 .2
C 4 A 0 .2 F 1 .8
1000
Intensities (counts)

800
600
400
200
0
10
20
30
40
2-Theta (°)
Brownmillerite Ca2 Alx Fe2-x O5
50
60
Problems with intensity ratios
Peak overlap – can exaggerate intensity or obscure peak (not fully
resolved)
 Particular problem with clinker/cement
C o u n ts
2000
4000
3000
2000
A li t e
B e li t e
A lu m i n a t e
F e r r i te
1500
Intensity(Counts)

1000
1000
500
0
30
0
29
30
35
31
32
40
P o s i ti o n [° 2 T h e ta ]
33
2-Theta(°)
34
35
36
37
Problems with intensity ratios

Crystallite size (not necessarily same as particle size)
 Large crystallites (i.e., thousands of unit cells) will produce sharp,
very intense diffraction peaks only at the precise location of the Bragg
angle (due to cancelling of diffractions by incoherent scattering).
 Small crystallites (~1 µm) will produce broad peaks due to incoherent
scattering at angles close to the Bragg angle
 Large crystals increase microabsorption
 <45 µm recommended, 2-10 µm optimal size, uniform size*, over
grinding can cause amorphous layer
Counts
200
Quartz large crystals
Quartz small crystals
100
0
20
25
Position [°2Theta]
30
Problems with intensity ratios

Microabsorption
 Strongly absorbing minerals (e.g., C4AF) will have reduced intensities
 Weakly absorbing minerals (e.g., periclase) will have greater than
average intensities.
 Reduced by fine grinding
 Not really a problem with internal standard approach
Problems with intensity ratios

Strain in crystal lattice
 Non-uniform microstrain (due to dislocations, vacancies, impurities,
etc.) results in peak broadening and possibly asymmetry
Other errors:



Flat specimen error
 For correct diffraction geometry, sample should be curved and lie
on focusing circle; flat sample causes an asymmetric peak
broadening at towards lower 2θ angles
Fixed slit (vs. variable slit)
 Larger area of sample irradiated at low 2θ values have less depth
of penetration; at higher 2θ, irradiated area is smaller, but depth of
penetration greater. These tend to offset, so get constant volume
being irradiated.
Axial divergence
 Occurs if X-ray beam diverges out of the plane of the focusing
circle; causes peak asymmetry at low angles; soller slits and curved
crystal monochromator limit divergence
Whole-pattern fitting (i.e., Rietveld analysis)

Rietveld (1969) developed a method to refine crystal structure
information using neutron powder diffraction
 Uses ‘initial’ crystal structure as a starting point to calculate the
expected diffraction profile based on physics
 Differences between the calculated profile and the measured profile
are minimised by a least squares iterative approach
 Uses all peaks and the complete profile (i.e., peak position, peak
intensity, and peak shape) in the analysis
 Many variables (errors) can be accounted for (e.g., preferred
orientation, crystallite size, strain, peak overlap, peak asymmetry,
site occupancy, absorption, diffraction geometry, etc.)
 pattern is sum total of all of the effects
 Can be used for standardless quantification of polycrystalline
materials
 Capable of much greater accuracy than intensity ratio methods
Rietveld analysis




How does it work?
Need to already know what is in the mixture (phase identification)
Need solved unit cell structural information for each compound in
mixture
 Unit cell parameters (dimensions, angles, crystal system, space
group)
 Atom type and positions (and site occupancy)
Inorganic Crystal Structure Database (ICSD)
 Maintained by FIZ Karlsruhe and NIST
 Contains 89,064 entries of inorganic crystal structure data (does
not contain everything you may want)
ICSD database
Calcite 80869
ICSD database

Super cell for alite (C3S) contains 226 atoms


Very computer intensive
4320 calculated peaks contribute to final
profile
Rietveld analysis benefits

Rietveld analysis has advantages over conventional Bogue calculations
based on XRF data
 Bogue assumes pure compounds (know this is not true)
 Gives more accurate determination of major clinker phases
 underestimates alite (up to 20%), overestimates belite (up to 10%)
(Stutzman, 2004; Glasser, 2004), underestimates ferrite by ~2-3% (for
high Fe2O3 clinker) (Feldman et al., 2005)
Oxide
Polymorphism
– can detect and quantify different polymorphs of
wt. %
CaO
63.75 (e.g., cubic C A, orthorhombic C A)
compounds
3
3
SiO2
20.98
Al20Can
detect
4.90 and quantify other phases (e.g., free lime, periclase,
3
Fe2calcium
03
3.44
sulphates (different forms), alkali sulphates, etc.)
MgO
K 2O
Na2O
SO3
P2O5
LOI
Free lime
0.76
0.34
0.08
3.08
0.09
C 3S
C 2S
C 3A
C4AF
53.5
19.8
7.2
10.5
1.44
What about amorphous materials?
Rietveld gives a ‘normalised’ fit – assumes that everything is crystalline
and sum of mass fractions add up to 1 (or 100%)
Can use internal standard (all peaks) to calibrate scale factors
 Difference between calculated and actual is due to amorphous
content


C o u n ts
1000
Blast-furnace slag
A k e r m a n i te 0 .2 %
A k e r m a n i t e - g e h le n i t e ( 5 0 % A k ) 1 . 5 %
A k e r m a n i t e - g e h le n i t e ( 7 5 % A k ) 0 . 7 %
G e h le n i t e 0 . 4 %
C a lc i t e 0 . 3 %
M e r w i n i te 1 .1 %
A m o r p h o u s 9 5 .8 %
500
0
25
30
35
40
P o s i ti o n [° 2 T h e ta ]
e.g., Determination of glass content of blast-furnace slag