Quantitative Phase Analysis by X-ray Diffraction · The internal standard method The ratio of two...

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