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Quantitative Phase Analysis by X-ray Diffraction
Robert L. SnyderSchool of Materials Science and Engineering
Georgia Institute of Technology
Denver X-ray Conference 2009
Outline of Talk
• Vegard’s Law Analysis• Spiking or Method of Standard Additions• Absorption Diffraction Method• Internal Standard Method• I/Ic and the RIR• The Generalized RIR Method• Normalized or “Standardless” Analysis• Whole Pattern Fitting and Rietveld Analysis
Vegard’s Law Analysis
Quantitative Analysis using the Lattice parameter
Substitutional Solid Solution
Retained Austenite Analysis
Deviations from Vegard’s Law
Quantitative Analysis using Line Intensities
• First done by L. Navias (GE) in 1925 on sillimanite and mullite.
• Today typical accuracy worse than 10%• Best RIR analysis ~3%• Best multi-line Copland-Bragg Analysis
~1%• Routine Rietveld Analysis ~1%
Factors affecting line intensities
Table 13.1
Selection of Background locations
Figure 13.2
Trace of the Si(111) peak using Cr radiation – Tails > 1 degree each
Figure 13.3
Variation in RIR as a function of scan width
Figure 13.4
Types of line measured in quantitative analysis: Peak Height, Peak Area, Overlapped
Figure 13.5
Foundation of quantitative phase analysis
Equation 13.1
( )
23 2 22 20( )( 2 2 2 2)
1 cos 2 cos 264 sin cos
hk mhkl ahk
e
s
M XI eI Fr m c V
α
αα
αλ θ θπ θ θ μρ
ρ
⎡ ⎤⎢ ⎥⎡ ⎤ ⎡ ⎤⎛ ⎞ ⎛ ⎞+ ⎢ ⎥⎢ ⎥= ⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎢ ⎥⎛ ⎞⎢ ⎥ ⎢ ⎥⎝ ⎠⎝ ⎠ ⎣ ⎦⎣ ⎦ ⎢ ⎥⎜ ⎟
⎝ ⎠⎣ ⎦
l
l
Equation 13.2
For line hkl of phase alpha
( )( )
hkIe hk
s
K K Xα
αα
αμρ ρ
⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠
=l
l
( )
23 2 22 20( )( 2 2 2 2)
1 cos 2 cos 264 sin cos
hk mhkl ahk
e
s
M XI eI Fr m c V
α
αα
αλ θ θπ θ θ μρ
ρ
⎡ ⎤⎢ ⎥⎡ ⎤ ⎡ ⎤⎛ ⎞ ⎛ ⎞+ ⎢ ⎥⎢ ⎥= ⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎢ ⎥⎛ ⎞⎢ ⎥ ⎢ ⎥⎝ ⎠⎝ ⎠ ⎣ ⎦⎣ ⎦ ⎢ ⎥⎜ ⎟
⎝ ⎠⎣ ⎦
l
l
Kexperiment Kspecimen
Equation 13.3
The Problem: X is not an independent variable. All wt. fractions must be known to compute the mass absorption coefficient.
1
n
ii i
Xs
μρ
μρ
⎛ ⎞⎜ ⎟⎜ ⎟ =⎜ ⎟
=⎝ ⎠
⎛ ⎞⎜ ⎟⎝ ⎠
∑
For a specimen containing n elements:
Nonlinearity due to matrix absorption
Compare Ihkl of a line in a mixture to its value in pure alpha
The case of polymorphs
( )0( )
hk
hk
IX
Iα
αα
=l
l
Zirconia oxygen sensor
High Mg stabilized sensor
Worry about assumptions in peak fitting!
Use of Klug’s Equation for any two phase mixture
Equation 13.8
( )( ) [ ( / ) ( / ) ]
e hkhk
K K XI
X Xα α
αα α α β βρ μ ρ μ ρ
=+l
l
( )
( )0 1( / )hk
e hkK KI
α
α
α αρ μ ρ=
l
l
For the pure α
phase:
Equation 13.10
Comparison of any I to that in the pure phase
( )
( )0
( / )( / ) ( / )
hk
hkI XI X X
α
α α α
α α β β
μ ρμ ρ μ ρ
=+
l
l
But, since
X + X = 1
Equation 13.11
Equation 13.12
Then
( )
( )0
( / )[( / ) ( / ) ] ( / )
hk
hkI XI X
α
α α α
α α β β
μ ρμ ρ μ ρ μ ρ
=− +
l
l
( )
( )
( )0
( )0
( / )
( / ) [( / ) ( / ) ]
hk
hk
hk
hk
II
X II
α
α
αα
αα
α α β
μ ρ
μ ρ μ ρ μ ρ=
− −
l
l
l
l
This equation can be used to plot a standard curve of I/I0
vs XαIt can also be rearranged to give Klug’s equation:
Use of measured mass attenuation coefficients
Figure 13.7
Use of mass absorption coef’s derived from elemental chemical analysis data
Shale Analysis using Spray- dried specimens and
standardsPhase Prepared Measuredillite 47.2% 45.1(4.1)quartz 34.1% 31.1(1.9)feldspar 11.4% 11.6(1.7)chlorite 7.3% 7.4(0.6)
The Basis of the Method of Standard Additions
The ratio of a line from phase to a line from any phase in aspecimen causes the mass absorption coefficient to cancel!
( )( ) ( / )
e hkhk
s
K K XI α α
ααρ μ ρ
= l
l
( ) '( ) ' ( / )
e hkhk
s
K K XI β β
ββρ μ ρ
= l
l
Method of standard additions or the Spiking Method
Equation 13.15
( ) ( )
( ) ' ( ) '
( )( )( )hk hk
hk hk
I K XI K X
α α β α
β β α β
ρρ
=l l
l l
If we add Y grams of pure phase to the original specimen
Equation 13.16
( ) ( )
( ) ( ) '
( )hkl hk
hkl hk
I K X YI K X
α α β α α
β β α β
ρρ′
+= l
l
Where Xβ
is the original weight fraction of the reference phase
Equation 13.17
I(hkl )α
I(hkl ′ ) β
= K (Xα + Yα )
Thus we have a linear equation
Spiking Method
Iα/Iref
Yα
Xα
Yα
= grams of α added per gram of original specimen Xα
= concentration of α in original specimen
The Basis of the Internal Standard Method
The ratio of a line from phase to a line from phase in anyspecimen causes the mass absorption coefficient to cancel!
( )( ) ( / )
e hkhk
s
K K XI α α
ααρ μ ρ
= l
l
( ) '( ) ' ( / )
e hkhk
s
K K XI β β
ββρ μ ρ
= l
l
The internal standard method The ratio of two I’s is a direct measure of the wt. ratio
I(hkl)α
I(hkl ′ ) β
= kXα
XβEquation 13.18
Thus, adding a phase of known concentration to the specimenwill permit the evaluation of k for a known and then the evaluation of the wt. fraction of alpha in any unknown
Internal Standard Method
I/Icorundum
• The slope of the Internal Standard curve is a materials constant.
• DeWolf and Visser (1966) proposed that all materials be mixed 50:50 with corundum and the ratio of the 100% lines be published with reference patterns.
• ICDD has a great number in the current PDF
Generalized Reference Intensity Ratio
Equation 13.19
RIRα ,β =I(hkl)α
I(hkl ′ ) β
⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟
I( hkl ′ ) β
rel
I(hkl )α
rel
⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟
Xβ
Xα
⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟
Quantitative analysis with RIR’sEquation 13.20
Equation 13.21
Equation 13.22
Xα =I(hkl )α
I(hkl ′ ) β
⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟
I( hkl ′ ) β
rel
I(hkl )α
rel
⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟
Xβ
RIRα ,β
⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟
Xα =I(hkl )α
I(hkl ′ ) β
⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟
I( hkl ′ ) β
rel
I(hkl )α
rel
⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟
RIRβ ,c
RIRα ,c
Xβ
⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟
RIRα ,β =RIRα ,γ
RIRβ,γ
The Normalized RIR method, Chung Method, “Standardless Method”, Matrix flushing method, etc.
Equation 13.23
Equation 13.24
Equation 13.25
Xα
Xβ
=I(hkl )α
I(hkl ′ ) β
⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟
I( hkl ′ ) β
rel
I(hkl )α
rel
⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟
RIRβ ,c
RIRα ,c
⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟
X j =1j =1
n
∑
Xα =I(hkl )α
RIRα I(hkl )α
rel1
I(hkl )' j RIRj I( hkl ′ ) j
rel( )j=1
No.of phases∑
⎡
⎣
⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥
Constrained XRD phase analysis Generalized internal standard method
Equation 13.26
In
I(hkl)'std
=I(hkl)1
rel
I(hkl)' stdrel RIR1,std
⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟
X1
Xstd
+I(hkl )2
rel
I(hkl)' stdrel RIR2,std
⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟
X2
Xstd
+...+I(hkl) j
rel
I(hkl )'stdrel RIR j,std
⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟
Xj
Xstd
+ ε
Copland Bragg Analysis
Copeland-Bragg Analysis
Spray-dried fly-ash specimens and standards with and without use of overlapped lines
Phase Prepared OnlyRes.Lines
Found AllLines
Found
Mullite 0.50 1 0.530(20) 6 0.506(22)quartz 0.15 1 0.171(10) 4 0.166(7)hematite 0.10 2 0.101(3) 5 0.104(5)glass 0.25 - 0.198(23) - 0.224(24)
Quantitative analysis using the calculated diffraction pattern
Figure 13.9
Use of total pattern Rietveld Quant
Equation 13.27
Equation 13.28
Equation 13.29
R = wj I j (0) − I j(c )j
∑ 2
I j(c ) = Sα K(hkl)α G(Δθ j ,(hkl )α )P(hkl) + Ib(c )(hkl)∑
Sα =I(hkl )α
K(hkl )α
Equation 13.30
Equation 13.31
Equation 13.32
The Rietveld scale factor contains the wt. fraction of each phase
For a pure phase:
For a mixture:
Comparing to our fundamental I eq.we can substitute tosolve for S.
Equation 13.33
The Rietveld method with an internal standard
Equation 13.34
Normalized internal standard analysis using Rietveld
Equation 13.35
Equation 13.36
Equation 13.37
Equation 13.38
Normalization equation only valid when amorphous phases are absent
Rietveld Quantitative Analysis 93% sanidine, 7% albite, R=26%
Cordierite
Auto Catalyst Raw Data
Tetragonally Stabilized Zirconia~ 100 A Crystallites
Zirconia simulations25 A
Experimental100 A250 A1000 A
Rietveld Analysis – PANalytical HighScore Plus
75% Cordierite, 25% Stabilized ZirconiaExperimentally Refined Cell Parameters
Full pattern fitting with experimental patterns
Equation 13.39
Detection of low concentrations
Figure 13.11
Sigmas are for 95% confidence (i.e. 2 )
Palaboora South Africa
Phosphate mine
Tubing carrying phosphate slurry
Apatite slurry in real-time XRD
Real-time analysis fed back to control surfactant concentration
