Quantitative Phase Analysis · 2015-08-03 · Quantitative Phase Analysis Theory and Practice Ian C. Madsen CSIRO Mineral Resources Flagship QUANTITATIVE PHASE ANALYSIS WORKSHOP –

Quantitative Phase AnalysisTheory and PracticeTheory and PracticeIan C. Madsen

CSIRO Mineral Resources Flagship

QUANTITATIVE PHASE ANALYSIS WORKSHOP – DXC2015, WESTMINSTER, COLORADO, USA

Private Bag 10, Clayton South 3169, Victoria, [email protected]

Q , , ,

Outline

• Basis of quantitative phase1 analysis (QPA)o Single peak methodso Whole pattern methods

• H t l t hi h th d t• How to select which method to use

• Some lecture notes available at :• Some lecture notes available at :‐o http://www.crystalerice.org/Erice2011/books/Book_Erice2011_PD.pdf

1 Phase = a crystallographically distinct component of the sample not to be confused with the “phase problem” in structure solution

Ian Madsen | TOPAS Training – Level 1 2013 |2 | Ian Madsen | TOPAS Training Course – Level 1 | 2013 |2 | Ian Madsen | CSIRO Mineral Resources Flagship | Quantitative Phase Analysis | DXC2015, Westminster, Colorado, USA2 |

not to be confused with the phase problem in structure solution

Sources of Information for Quantitative XRDWide coverage of general principlesWide coverage of general principles

o Zevin & Kimmel (1995) Q i i X Diff– Quantitative X‐ray Diffractometry.

– Springer New York


An Opportunity for Shameless Self Citation

• Chapter 10o Quantitative Phase Analysis Using

the Rietveld Method– I.C. Madsen, N.V.Y. Scarlett,, ,

D.P. Riley and M.D. Raven


International Tables for CrystallographyVolume H – Powder Diffraction

• Contents (provisional):o Part 1 Introduction

Volume H – Powder Diffraction

o Part 1. Introductiono Part 2. Instrumentation and sample

preparationo Part 3. Methodologygyo Part 4. Structure determinationo Part 5. Defects, texture,

microstructure and fibresP 6 S fo Part 6. Software

o Part 7. Applications• Volume H will be the key

reference for all powderreference for all powder diffractionists from beginners to advanced practitioners

• Scheduled publication date: o 2016


Analytical Methods of Phase AnalysisIndirect methodsDirect methodsDirect methodsXRD for quantitative phase analysisRange of complexity of materials analysed

Analytical Methods of Phase AnalysisIndirect methods

•Measure, for example, bulk chemistry

Indirect methods

o Apportion elemental abundances according to assumed composition of each phase normative calculation – Bogue method for Portland cementg

• Potential for error in QPA if errors exist in assumed compositions• Instability in method when phases have similar chemical composition

• Not applicable when phases have identical chemical composition P l h / il h i / h io Polymorphs – anatase/rutile, hematite/maghemite


Analytical Methods of Phase AnalysisDirect methods

•Magnetic susceptibility

Direct methods

o Limited to samples with magnetic phases• Selective dissolution

o Rate of dissolution can be phase dependanto Rate of dissolution can be phase dependant• Density measurements

o Physical separation of phaseso Physical separation of phases• Image analysis

o Optical & e‐beam images – issues with stereology• Thermal analysis

o Magnitude of endo‐/exo‐thermic features during phase transitions relate to phase contentphase content

• Diffraction based methods ………………………….


XRD for Quantitative Phase Analysis

• “Quantitative phase analysis by X‐ray diffraction (QXRD) is the l l ti l t h i th t i t l h iti ” 1only analytical technique that is truly phase sensitive” 1

o Diffraction data derived directly from the crystal structure of each phase– Results are not inferred via indirect measurementResults are not inferred via indirect measurement

o Capable of analysing polymorphs

• Mathematical basis of QPA is well established, butQ ,o Limitations on accuracy are mostly experimental

• Many sources of error yo Instrument configuration Particle statisticso Counting error Preferred orientationo Microabsorptiono Operator error ! PICNIC Problem In Chair, Not In Computer


1 Chung & Smith (2000)

Range of Complexity in QPASynthetic materials

• Sample 1G from IUCr CPD round robin

Synthetic materials

o ‘Simple’ – 3 well defined phases with high symmetry, small unit cells– Little peak overlap

Corundum 33.08 %Fl it 33 56 %

150Fluorite 33.56 %Zincite 33.35 %

qrt(C

ount

s) 100

Sq

50

2Th Degrees7065605550454035302520


Range of Complexity in QPA Mineralogical samples

• Mineral samples are complex • QPA of mineral samples is rarely a straight forward exercise !

Mineralogical samples

o Multi‐phase (20 not uncommon) o Inhomogeneous at all size ranges• Sample related issues

straight‐forward exercise !• Difficult to standardize methodology

115

110

105

100

95

90

Pyrite 16.41 %Pentlandite 54.56 %Magnetite 10.73 %Magnesite 4.31 %Violerite 1.91 %Galena 0.06 %Millerite 0.33 %Pyrrhotite1 1.42 %

• Sample related issueso Poorly crystallinity – Clays, goethite, nontronite

Nickel Concentrate

85

80

75

70

65

60

55

Pyrrhotite2 1.24 %Talc 3.89 %Hydrotalcite 0.63 %Nepouite 4.51 %o Variable chemical composition

(solid solution) of phaseso Preferred orientation 55

50

45

40

35

30

25

o Preferred orientation, micro‐absorption etc..

2Theta (deg)7570656055504540353025201510

20

15

10

5

0


2Theta (deg)

Basis of Quantitative Phase AnalysisSingle peak methodsWhole pattern methodsWhole pattern methods

Commonly Used Diffraction Based QPA Methods Divided Into Two Distinct Groups• Traditional ‘single peak’ methods

Divided Into Two Distinct Groups

o Rely on measurement of intensity of a peak, or group of peaks, for each phase of interest

o Assumes intensity is representative of phase concentrationy p po Affected by peak overlap, preferred orientation and microabsorption•Whole pattern methods

o Compare wide 2θ range diffraction data with a calculated pattern– Summation of individual phase components which have either been:‐

• Measured from pure phase samples• Measured from pure phase samples• Calculated from crystal structure information


Complex Relationship Between Intensity of Diffracted Peaks & Phase Concentration

22243 i221 WMI *

2

2

222

)(242

430

)(sin2

cossin2cos2cos1

232

m

mhkl

hkl

ehkl

WBExpFV

Mcm

er

II

• Experiment / Instrument Dependanto I (hkl)α = Intensity of reflection of hkl in

phase αI0 = incident beam intensity

• Phase Dependanto Mhkl = multiplicity of reflection hkl of phase α

Vα = volume of unit cell phase αF(hkl)α= structure factor ‐ reflection hkl of 0 y

r = distance from specimen to detectorλ = X‐ray wavelength(e2/mec2) = square of classical electron radius

(hkl)αphase α2θ = diffraction angle of reflection hkl of phase αρα = density of phase αB t i di l t (th l)o 2θm = diffraction angle of the

monochromator• Sample Dependant

W weight fraction of phase α in sample

B = atomic displacement (thermal) parameter

o Wα = weight fraction of phase α in sampleμm* = mass absorption coefficient (MAC) of the entire sample


Single Peak MethodsAbsorption‐diffraction methodInternal standard methodInternal standard methodHow to choose an internal standard

R f i t it tiReference intensity ratioMatrix flushing method

Single Peak MethodsAbsorption Diffraction method

o Rearrange to extract phase abundances in real samples

Absorption‐Diffraction method

W

i h f i f h

abundances in real samples*m

iiWCI

I *o Wα = weight fraction of phase αo Iiα = intensity of peak (or group of

peaks) i for phase α

i

mi

CIW

o Ciα = calibration constanto μm* = whole sample MAC

R t d i C f

o Need to determine μm* for each sample and standard

o Need to prepare standard(s)o Rearrange to derive Ciα from standard(s) with known Wα

o Need to prepare standard(s)

I *

W

IC mii


Determining Sample Mass Absorption Coefficient

Direct Measurement• M b i i i λ

Calculation• C l l * f h f h• Measure beam intensity using same λ

used in XRD data collectiono Sample of known thickness t

• Calculate μm* from the sum of the products of :‐

o Theoretical MAC (j) of each element (or phase)• I = intensity with sample in

• I0 = intensity with sample out

phase) o Weight fractions (Wj) of all n elements (or phases) in the sample

tII

mm*exp j

n

jm W ** I0

jj

j1

t

– Note – using chemical analysis is likely to be more accurate since amorphous content (not analysed by XRD) is included in the calculation

Io I

μρ

λ


Single Peak MethodsInternal standard method

• Recall relationship for phase α • Divide two equations

Internal standard method

o Eliminate μm*

*iiWCI

iisi CCWI

• Include an internal standard sin known quantityW and

m

ijs

js

is

js

i CCWI

in known quantity Ws and measure peak(s) intensity Ijs

o where Cjsiα is a calibration constant specific to phase, standard and lines used

•Wα determined by :‐*m

sjsjs

WCI

iIWW m

js

iijs

s

II

CWW


How to Choose an Internal Standard

• Material should be stable & unreactiveo Especially for in situ studies

• Simple diffraction pattern – minimal overlap with sample peaks

• Standard MAC should be similar to sample MACo Avoid introducing microabsorption effects

• Minimal sample related effect on observed intensitieso No preferred orientation • 100% (or known) crystallinity

Mi i l ‘ i i ’o Minimal ‘graininess’


How to Choose an Internal Standard (cont’d)

• Possibilitieso ‐Al2O3 (corundum) TiO2 (rutile) ZnO (zincite) Cr2O3 (eskolaite)

‐Fe2O3 (hematite) CeO2 (cerianite) CaF2 (fluorite) C (diamond)

• Alternate approacho Use an independent measure (e.g. chemical analysis) to derive the

concentration of a phase already present in the sampleo Designate it as the internal standard


Single Peak MethodsReference Intensity Ratio RIR

• An instrument independent phase constant

Reference Intensity Ratio RIR

•Developed specifically for QPA• Ratio of :‐

k f h do Strongest peak of phase , ando Strongest peak of standard s

sijs

si RIRCWW

II

j

js WI


Single Peak MethodsReference Intensity Ratio RIR (cont’d)

• If strongest peaks are not available

Reference Intensity Ratio RIR (cont d)

o Use weaker peaks & scale relative intensities – can keep same RIR value– Irel = ratio of intensity of peak used to most intense peak for phase

ss

reli

reljs

js

i RIRWW

II

II

•Most common standard is corundum (α‐Al2O3)RIR I/I

ijs

• RIR equates to I/Ico “I over Icorundum”o These are the most commonly reported values in the literatureo These are the most commonly reported values in the literature



• Collated lists of RIR values for common phases in ICDD database


• Take care in selection of RIR value for a particular experiment • RIR depends upon the data collection and measurement strategy

k h i h i d k h l l ho Peak height, integrated peak area, whole pattern, X‐ray wavelengtho Must match the conditions used in analysis• Do not rely on published values ‐ determine for current materialDo not rely on published values determine for current material



• RIR CaF2 Fluorite


o ICDD (various) Range 3.84 ‐ 4.14 Outlier 2.40 – Mean (n=11) 3.89(10) Relative error 2.7%

o Madsen * Calc 3 67 Measured 3 629o Madsen Calc 3.67 Measured 3.629

• RIR ZnO Zinciteo ICDD (various) Range 4.85 – 5.87 Outlier 4.50– Mean (n=20) 5.32(19 ) Relative error 3.6%M d * C l 4 90 M d 4 943o Madsen * Calc 4.90 Measured 4.943

• If published values of RIR are used• If published values of RIR are used o Then QPA should be considered as only semi‐quantitative


* Calculated patterns generated in TOPAS. Measured values derived using Sample 1 from IUCr CPD QPA round robin

Single Peak MethodsMatrix Flushing Method *

• An important feature of RIR based techniques is

Matrix Flushing Method *

o Once the RIRs are determined for the analyte phases of interest,standard phase does not need to be present in the sample

• Remove effect of μm* by taking ratio of intensity of phase α andRemove effect of μm by taking ratio of intensity of phase α and another unknown phase β

rel RIRIIW

s

sreli

j

j

i

RIRRIR

II

II

WW


* Chung (1974), J.Appl.Cryst, 7, 519-525 and Chung (1974), J.Appl.Cryst, 7, 526-531.

Single Peak MethodsMatrix Flushing Method (cont’d)

• If all components are crystalline and included in the analysis, i t d dditi l t i t

Matrix Flushing Method (cont d)

introduce an additional constraint

n

0.11

k

kW

• Sum of all weight fractions = unity (or known value)


Single Peak MethodsMatrix Flushing Method (cont’d)

• Forms a system of n linear equations

Matrix Flushing Method (cont d)

o Solve to derive weight fractions of all components in the analysis

1

nkII

1

krelkks

krel

s IRIRI

IRIRIW

o Weight fractions are correct relative to each other – May not be correct in an absolute sense

If unidentified or amorphous materials are present in the sample– If unidentified or amorphous materials are present in the sample, reported phase abundances may be overestimated

– Add internal standard, or use knowledge of the amount of a component h d t i d b th t h i t t t b l t b dphase determined by another technique, to extract absolute abundances


* Chung (1974), J.Appl.Cryst, 7, 519-525 and Chung (1974), J.Appl.Cryst, 7, 526-531.

Demonstration of methodsSingle peak methods

IUCr Commission on Powder Diffraction

90

1000

10 c

Round Robin on Quantitative Phase Analysis• Experimental design for Sample 1

Ei h i f 3 h

%) 70

80

90

20

30

o Eight mixtures of 3 phases– Corundum – ‐Al2O3

– Fluorite – CaF2

ndum

(wt%

50

60

70 Zincite (w

40

50 f

– Zincite – ZnOo Each phase present at a range of

concentrations

Coru

nd

30

40

50 wt%)60

70

d

go ~ 1.5, 5, 15, 30, 55, 95 wt%• Data collection

o 3 replicates

10

20

30

80

90

de

ho 3 replicates

0 10 20 30 40 50 60 70 80 90 100Fluorite (wt%)

0

10

100 ab


( )

Weight Fractions of Phases in IUCr CPD Sample 1

Sample Corundum Fluorite Zincite1A 0.0115 0.9481 0.04041B 0.9431 0.0433 0.01361C 0 0504 0 0136 0 93591C 0.0504 0.0136 0.93591D 0.1353 0.5358 0.32891E 0.5512 0.2962 0.15251F 0.2706 0.1772 0.55221G 0.3137 0.3442 0.34211H 0 3512 0 3469 0 30191H 0.3512 0.3469 0.3019


Nett Peak Intensity* Derived Using Profile Fitting Corundum (113) Fluorite (022) & Zincite (011)

Sample Corundum Fluorite Zincite μm*

Corundum (113), Fluorite (022) & Zincite (011)

1A 34.8 (0.6) 8958.7 (33.0) 509.9 (6.0) 93.021B 6561.3 (28.6) 1095.5 (7.1) 474.3 (3.8) 34.451C 244 4 (0 9) 250 9 (10 1) 22898 0 (37 0) 49 031C 244.4 (0.9) 250.9 (10.1) 22898.0 (37.0) 49.031D 474.5 (3.5) 6559.6 (2.8) 5468.5 (9.5) 71.711E 2525.3 (27.9) 4835.5 (27.0) 3370.7 (16.3) 53.171F 1251.3 (7.8) 2935.8 (9.0) 12494.9 (22.4) 52.671G 1295.0 (8.7) 5041.7 (17.0) 6787.9 (26.6) 59.641H 1436 5 (7 3) 5132 0 (13 6) 5996 8 (59 5) 59 101H 1436.5 (7.3) 5132.0 (13.6) 5996.8 (59.5) 59.10

* Values are average of 3 replicatesμm* calculated from chemical (XRF) analysis


Demonstration of MethodsAbsorption – diffraction method

•Determine calibration constant for fluorite using Sample 1D

Absorption – diffraction method

919,87753580

71.716.6559*

,, W

IC mii

• A l t S l 1H

5358.0W

• Apply to Sample 1H

weigh)0.3469 (cf 3455.0877919

1.590.5132*

, m

i CIW

877919,,

iC


Demonstration of MethodsAbsorption – diffraction methodAbsorption – diffraction method

2100

1

2

80

100

%)

Fluorite

1

60

wt%

)

ntratio

n (w

t%

0

40 Bias (w

ysed

Con

cen

‐120An

aly

‐200 20 40 60 80 100

Known Concentration (wt%)


Analysed wt% (closed diamond) – left axis Bias = Analysed – Weighed (open triangle) – right axis

Demonstration of MethodsInternal standard method

•Determine calibration constant for fluorite using Sample 1H

Internal standard method

o Designate zincite as internal standard

3019005132fl i WI7448.0

3469.03019.0

8.59960.5132

ijs

fluorite

zincite

zincite

fluorite CWW

II

• Apply to Sample 1DApply to Sample 1D

i h d)0 5358( f529706.65593289.0fluoritezincite IWW weighed)0.5358(cf 5297.05.54687448.0

zincite

fluoriteijs

zincitefluorite IC

W


Demonstration of MethodsInternal standard methodInternal standard method

2100

1

2

80

90

100

%)

Fluorite

1

60

70

wt%

)

ntratio

n (w

t%

0

30

40

50

Bias (w

ysed

Con

cen

‐1

10

20

30

Analy

‐200 20 40 60 80 100




Demonstration of MethodsDetermination of RIR

• Calculate RIR for fluorite and zincite using Sample 1H

Determination of RIR

W

WI

IRIR corundumWIcorundum

617.3346903512.0

514360.5132

fluoriteRIR3469.05.1436

3512085996 856.43019.03512.0

5.14368.5996

zinciteRIR


Demonstration of MethodsDetermination of RIR

• 24 RIR determinations

Determination of RIR

o 8 samples, 3 replicates• Good agreement at intermediate concentrations

5.0

o

Zinciteintermediate concentrations

• Significant deviations at low corundum concentration

4.5

tensity

Ratio

o Insufficient intensity to ensure sufficient accuracy in the RIR

4.0

eferen

ce Int

3.5Re Fluorite

3.00 20 40 60 80 100

Corundum Concentration (wt%)


Demonstration of MethodsRIRmethod

• Calculate fluorite concentration in Sample 1D

RIRmethod

61736.6559fluorite

RIRI

weighed)0.5358 (cf 5312.0

856.45.5468

617.36.6559

0.15.474

617.3

1

n

k k

k

fluoritefluorite

RIRI

RIRW

1k ksRIR


Demonstration of MethodsRIRmethodRIRmethod

2 000 2.0

80

90

100

%)

Fluorite

1.0

60

70

t%)

tration (w

t%

0.0

30

40

50

Bias (w

t

sed Co

ncen

t

‐1.0

10

20

30

Analys

‐2.000 20 40 60 80 100




Whole Pattern MethodsPattern summationRietveld basedRietveld basedExternal standardInternal standardInternal standardMatrix flushing (Rietveld context)

Methods for Quantitative Phase AnalysisWhole pattern techniques

• Benefits in using whole patterns c.f. conventional single peak th d d i f

Whole pattern techniques

methods derive from :‐o The use of the entire diffraction pattern hundreds or

thousands of reflections contribute to the resulto Minimising the impact of some systematic sample

related effects such as preferred orientation and extinctiono The ability to accurately deconvolute overlapping peakso The ability to accurately deconvolute overlapping peaks– More complex patterns can be analysed


Methods for Quantitative Phase AnalysisFull pattern fitting techniques

• Based on the principle that an observed diffraction pattern is the f th i di id l t

Full pattern fitting techniques

sum of the individual components• Full‐pattern fitting methods

o GMquant D K Smitho GMquant – D.K. Smitho FULLPAT – S. J. Chipera and D. L. Bisho RockJock – D.D. Eberlo ??? – M. Batchelder and G. Cressey•Wide‐range diffraction patterns of (pure) phases of interest are

l d d d d ith th b d diff ti d tscaled, summed and compared with the observed diffraction datao Use least squares fitting to obtain best fit


Sample 1G ≈ Mix of Corundum, Fluorite, ZinciteZincite contributionZincite contribution165160155150150145140135130125120

unts

)

115110105100

959085

Sqr

t(Cou 85

807570656055504540353025

2Theta (deg)80757065605550454035302520

201510

50


( g)

Diffraction Pattern for Si FlourModelled via pattern summationModelled via pattern summation

80

75

70

65

unts

)

60

55

50

Sqr

t(Cou 45

40

35

3030

25

20

15

2Theta (deg)12115110105100959085807570656055504540353025201510

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80

75

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unts

)

60

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Sqr

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40

35

3030

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20

15

2Theta (deg)12115110105100959085807570656055504540353025201510

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80

75

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65

unts

)

60

55

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Sqr

t(Cou 45

40

35

3030

25

20

15

2Theta (deg)12115110105100959085807570656055504540353025201510

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5



80

75

70

65

unts

)

60

55

50

Sqr

t(Cou 45

40

35

3030

25

20

15

2Theta (deg)12115110105100959085807570656055504540353025201510

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5



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75

70

65

unts

)

60

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Sqr

t(Cou 45

40

35

3030

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15

2Theta (deg)12115110105100959085807570656055504540353025201510

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10

5


Methods for Quantitative Phase AnalysisFull pattern fitting techniques (cont’d)

• Relies on the generation of a library of standard patterns for each h t d i th l i

Full pattern fitting techniques (cont d)

phase expected in the analysis• Collect library patterns under the same instrumental conditions as those used in subsequent analysisthose used in subsequent analysis

• Selection of standards which match the phases in the unknown sample is a critically important step

o Phase composition (solid solution) changes may affect peak positionso Crystallite size / strain may affect peaks width & shape• Lib i f f• Library can consist of patterns of :‐

o Well‐ordered phaseso Less well ordered material such as glasses polymers clay mineralso Less well ordered material such as glasses, polymers, clay mineralso Calculated patterns


Methods for Quantitative Phase AnalysisFull pattern fitting techniques (cont’d)

•Weight fractions obtained using refined scale factors in :‐

Full pattern fitting techniques (cont d)

o Absorption‐diffraction method where sample MAC is calculated from analysis of chemical composition

o Internal standard method where scale factors are normalised to an internal standard, typically corundum, using a RIR approach


Chipera & Bish, 2002, Chipera & Bish, 2013

Relationship Between Rietveld Scale Factor and Phase Concentration

1WK *2

1

m

WVKS

o Where– S = Rietveld scale factor for phase

W weight fraction of phase – W = weight fraction of phase – = density of phase – m

* = mixture mass absorption coefficient (MAC)m

– V = volume of the unit cell for phase – K is an ‘experiment constant’ used to put W on an absolute basis

• This equation inherently contains the weight fraction information


Rietveld Based Techniques for QPAExtracting phase abundances

• Rearrange

Extracting phase abundances

KVSW m

*2

• But, phase density can be calculated from crystallographic

K

parameters

1 66054 ZM

•Where

1.66054

VWhere

o ZM = the mass of the unit cell contentso V = the unit cell volume

1.66054 = 1024 / 6.022 x 1023

converts in AMU/Å3 to g/cm3


Substitute for Density

K

ZMVSW m*

• K is an ‘experiment constant’ for the instrumental setup 1,2

K

p po Used to put W on an absolute basiso Dependant only on instrumental and data collection conditions

I d d f i di id l h d ll l l do Independent of individual phase and overall sample‐related parameterso A single measurement is (usually) sufficient to determine K

• ZMV becomes a dynamic ‘phase constant’ for phase • ZMV becomes a dynamic phase constant for phase o It is updated as the structure is refinedo Can be determined from published/refined crystal structure parametersp / y p

• Referred to hereafter as the External Standard Method


1 O‘Connor & Raven (1988) Powder Diffraction, 3(1), 2-6. – Rec‘d 31/03/1987 2 Bish & Howard (1988) J.Applied Crystallogr., 21, 86-91. – Rec‘d 30/03/1987

Rietveld Based Techniques for QPAIssues in the application of the external standard method

•Need to measure K and ensure that instrumental conditions do t h b t t f K d d t ll ti f

Issues in the application of the external standard method

not change between measurement of K and data collection from samples

• Need to measure or calculate μ *Need to measure or calculate μmo Difficult to measure directlyo Need total chemistry or QPA for calculation• Can eliminate the need to know K and μm* by:‐

o Adding a known amount Ws of a well characterised standard S to the sample

ZMVSW mss*

K

Ws


Eliminating K and μm*

• Divide equation for phase by equation for standard S,

*

*m

ZMVSK

KZMVS

WW

msss ZMVSKW

• Rearrange , eliminate K, µm*

s ZMVSZMVSWW

• Effect of sample MAC & experiment conditions are eliminated

ss ZMVS

p p• Referred to hereafter as the Internal Standard Method


Advantages in Internal and External Standard Approach to QPA• Within the limits of experimental accuracy, the internal and

t l t d d h d b l t hexternal standard approaches produce absolute phase abundances

• Possible to estimate the amount of amorphous / non determined• Possible to estimate the amount of amorphous / non‐determined material Wunknown in the sample

o Equals the difference between unity & sum of the (absolute) analysed phase q y ( ) y pabundances

n

jabsolutejunknown WW

1)()( 0.1

j


Matrix Flushing Method (MFM) 1, 2Rietveld Context (ZMVmethod)

•MFM applies an additional constraint

Rietveld Context (ZMVmethod)

o Sum of analysed weight fractions = 1.0• Put MFM into Rietveld context 3,4

o Weight fraction of phase in an n phase system is :o Weight fraction of phase in an n phase system is :‐

)(

ZMVSW

1)(

n

jjj ZMVS

W

o Where– S = the Rietveld scale factor

ZM f i ll

1j

– ZM = mass of unit cell contents– V = unit cell volume 1 Chung (1974a) 2 Chung (1974b)

3 Hill & Howard (1987) J.Appl. Cryst., 20, 467-474. – Rec‘d 02/04/19874 Bish & Howard (1988) J.Appl. Cryst., 21, 86-91. – Rec‘d 30/03/1987


Matrix Flushing Method (MFM) 1, 2Rietveld Context (cont’d)

• This approach is the most widely used for Rietveld based QPA

Rietveld Context (cont d)

o Almost universally coded into Rietveld analysis programso This is probably the default QPA reported• BUT only produces relative phase abundances• BUT, only produces relative phase abundances.

o If the sample contains amorphous phases and/or unidentified crystalline phases analysed weight fractions will be overestimated

• If absolute abundances are required ..o Reaction kinetics via in situ studies

M t f h t to Measurement of amorphous contentetc... etc...

... then this method is not suitable !


Application of QPA MethodologyHow to select which QPA method to use

How to Select Which QPA Method to Use

• Experiment at the Australian Synchrotron by Webster et al. 1o Study of nucleation & crystal growth of gibbsite Al(OH)3– Context – Bayer process (extraction of Al from bauxite ores)

• S th ti B li (Al l d d ti l ti )• Synthetic Bayer liquors (Al‐loaded caustic solutions) o Seeded with various Fe‐oxides – in this example, goethite (‐FeOOH)– Use S‐XRD to follow mechanism & kinetics of phase formationUse S XRD to follow mechanism & kinetics of phase formation


1 Webster, N.A.S. et al, J. Appl. Cryst., 43: 466-472.

Experimental Configuration for Seeding Studies

• Sample environment1 l ill

Pressure line to 900psi

Quartz glass capillary (reaction vessel)o 1 mm quartz glass capillary

o Heated to 60 – 75°C using hot air blower

(reaction vessel)

o Slight pressure to prevent evaporation of fluid

• Simultaneous data collectiono Mythen multistrip detector,

Australian Synchrotrono 2 minutes per data set for ~3 hours• Rietveld based data analysis

o Three different QPA methods used to extract phase abundance at each

Thermocouple

extract phase abundance at each stage of the reaction Heater

(to 450°C)


Method #1QPA derived using ZMV (Hill & Howard) algorithm

• This is the most commonly used Rietveld based QPA methodologyR d hi ’ 100 %

QPA derived using ZMV (Hill & Howard) algorithm

• Reported goethite conc’n starts at 100wt%o Normalised to 100wt% as it is the only phase in the analysiso But, goethite added at 14.13wt% in total sample (solid + fluid)• Apparent decrease in goethite conc’n as Al(OH)3 polymorphs crystallise

o But, goethite will not dissolve or react in this environmento Total Al(OH) 35wt% 35o Total Al(OH)3 35wt%

– Exceeds known Al addition

• B h i f thit &25

30

35

90

100

entr

atio

n (w

t%)

entr

atio

n (w

t%)

Goethite Gibbsite B it• Behaviour of goethite &

Al(OH)3 phases is an artefact of the analysis

10

15

20

70

80

ted

Al(O

H) 3

Con

ce

ed G

oeth

ite C

once Bayerite

Nordstrandite

o We are not considering the entire sample – only thecrystalline components 0

5

60

70

0 20 40 60 80 100 120 140 160 180 200

Unc

orre

ct

Unc

orre

cte

Elapsed Time (minutes)



Method #2QPA derived using internal standard method

• Goethite seed added in known amount – 14.13wt%A h i d h d i i

QPA derived using internal standard method

o Assume that it does not change during experimento Correct the concentrations of other phases using internal standard equation

• Al(OH)3 phase conc’n now ~7.5wt% = ~ ½ of the known Al additiono In agreement with performance from independent estimates

814 2

• Must consider the entire sample (solid & liquid)

5

6

7

8

14.0

14.1

14.2

ratio

n (w

t%)

ratio

n (w

t%)

• Now have ability to derivereaction kinetics from

2

3

4

5

13.8

13.9

Al(O

H) 3

Con

cent

r

Goe

thite

Con

cent

Goethite Gibbsite

absolute QPA0

1

2

13.6

13.7

0 20 40 60 80 100 120 140 160 180 200

Cor

rect

ed

Spik

ed G


Bayerite Nordstrandite



Method #3QPA derived using external standard method

•Need to determine experimental constant K

QPA derived using external standard method

• K determined from analysis of first data seto Use goethite scale factor, ZMV and known addition (= 14.13wt%)o Ignore μ * sealed system chemistry therefore μ * will not changeo Ignore μm – sealed system – chemistry, therefore μm , will not change

• But – synchrotron beam current decays during data collectiony y go Instrument conditions have changed– Need to allow for what amounts to a change in K

i

mii I

IK

ZMVSW 0

*

o Where– I0 & Ii = monitor count (or beam current) at start & in data set i respectively

iIK


Method #3QPA derived using external standard method (cont’d)

• Absolute phase abundances derived

QPA derived using external standard method (cont d)

• Al(OH)3 conc’n ~same as internal standard estimates• Now observe a slight decrease in goethite QPA (<1% relative)

h ld b h ?o What could be the cause ?

814 2

5

6

7

8

14.0

14.1

14.2

atio

n (w

t%)

ratio

n (w

t%)

2

3

4

5

13.8

13.9

Al(O

H) 3

Con

cent

ra

Goe

thite

Con

cent

rGoethite

Gibbsite

0

1

2

13.6

13.7

0 20 40 60 80 100 120 140 160 180 200

Cor

rect

ed

Cor

rect

ed G


Bayerite

Nordstrandite



SEM Image of Goethite‐Seeded Gibbsite Crystallisation• Initial goethite seed has particle size of ~0.1 x 0.6 µm

• During growth, gibbsite envelopes goethite seedo Progressively ‘shields’ goethite from X‐ray beam

White = Goethite FeOOH seed

Grey = Gibbsite Al(OH)3 grain• Gibbsite particles ~10µm

o In general agreement with size calculated from decrease incalculated from decrease in observed intensity

*)( )(I t

o µ = linear absorption coefficient of

*

0

)( )( tExpI

t

o µ = linear absorption coefficient of gibbsite at selected wavelength

o t = thickness


* Beer‘s Law

Standardless Determination of the Phase Constant C

Issues in the Determination of Phase Constant C

• To determine phase constant C, we may need :‐o Pure sample of phase of interest ‐ accurately reflects form of phase in ‘real’

sampleso A multiphase sample in which the phase concentration is known by other p p p y

means (e.g. chemical analysis, point counting etc...)

• There may be insufficient sample available to risk ‘contaminating’ i i h i l d dit with an internal standard

• The addition of an internal standard may also introduce i b ti bl i th l it f ttmicroabsorption problems or increase the complexity of patterns

which are already highly overlapped


Standardless Determination of the Phase Constant C• An alternative approach1,2 to deriving C relies on having a suite of

l t b l d hi hsamples to be analysed which :‐o Have the same phases present in all samples, and o Exhibit a wide range of composition of these phases in various samples ino Exhibit a wide range of composition of these phases in various samples in

order to stabilise the analysis

1 Zevin, L. S. & Kimmel, G. (1995). Quantitative x-ray diffractometry. Springer-Verlag New York, Inc2 Knudsen, T. (1981). Quantitative x-ray diffraction analysis with qualitative control of calibration samples. X-Ray Spectrom. 10, 54-56.


( ) y y y

Standardless Determination of the Phase Constant C (cont’d)• Reconsider the relationship between weight fraction W and

b d i t it i th b ti diff ti th dobserved intensity in the absorption‐diffraction method

miIW*

• Assume all m phases in system are known & included in analysis

iC

W

o Introduce an additional constraint – sum of all W is unity (or a known value)

m

0.11

j

jW

mWWW ............0.1 21


Standardless Determination of the Phase Constant C (cont’d)• In a system of n samples containing m phases, express as a set of

i lt tisimultaneous equations

mIII *11

*112

*111011Sample

mCCC 21

............0.1 1 Sample


1 Zevin & Kimmel, 1995



mIII *11

*112

*111011Sample

mCCC 21

............0.1 1 Sample

III *22

*222

*221

m

m

CI

CI

CI 22

2

222

1

221 ............0.1 2 Sample




mIII *11

*112

*111011Sample

mCCC 21

............0.1 1 Sample

III *22

*222

*221

m

m

CI

CI

CI 22

2

222

1

221 ............0.1 2 Sample

” ” ” ”

III ***

m

nnmnnnn

CI

CI

CI

2

2

1

1 ............0.1 n Sample


Where μn* is the mass absorption coefficient of the nth sample

Standardless Determination of the Phase Constant C (cont’d)• The simultaneous equations can be expressed in matrix notation

CIII01

m

m

CC

IIIIII

..

..0.10.1

2

1

22221

11211

m

..........0.1222221

mnmnn CIII ..0.1 21

L = I x C


NOTE – x denotes a matrix multiplication operation

Standardless Determination of the Phase Constant C (cont’d)• The simultaneous equations can be expressed in matrix notation

ICL

• wherewhere o L is a column vector (dimensions 1 x n) containing the known (or assumed)

sum of weight fractions for each sampleo C is a column vector (dimensions 1 x m) containing the calibration constants

for each phaseo I is a rectangular matrix (dimensions n rows x m columns) containing the g g

measured intensities (or scale factors) for each phase– Multiplied by the sample MAC, or normalised to an internal standard


Standardless Determination of the Phase Constant C (cont’d)• A least squares solution can be calculated using simple matrix

i l ti th d 1manipulation methods1

LIIIC TT 1)(

• where

)(

o T is the matrix transpose functiono ‐1 is the matrix inverse function


1 Knudsen, T. (1981). X-Ray Spectrometry. 10, 54-56.

Demonstration of Zevin/Knudsen Method

• Use single peak intensities & um* in MS‐Excel to calculate phase t t S l 1 it f IUC CPD d bi QPAconstants – Sample 1 suite from IUCr CPD round robin on QPA

Phase C C/Ccorun RIR*

Corundum 241357.8 1.0 1.0

Fluorite 877413.6 3.635 3.617Fluorite 877413.6 3.635 3.617

Zincite 1188997.4 4.926 4.856


* See earlier slide on RIR methodology

Demonstration of Zevin/Knudsen MethodApply calibration constants to all 24 samplesApply calibration constants to all 24 samples

2.0

1.0

1.5

0.0

0.5

(wt%

)

‐0.5Bias

d

‐1.5

‐1.0 Corundum

Fluorite

Zincite

‐2.00 20 40 60 80 100

Phase Concentration (wt%)


Zevin/Knudsen MethodImportant notes

• Standard samples are not required

Important notes

o No need for prior calibrationo The system is self‐calibrating and has only relied on having

a wide range of concentrations of the analysed phasesg y p• Only need to measure

o Intensity of each phase in each sample .....– Single peak intensity, or whole pattern scale factor

o ..... multiplied by μm*, or normalised to an internal standard

• There must be more samples than phases to be analysed• There must be more samples than phases to be analysed


Ian MadsenIan Madsen Honorary Fellow – Diffraction Science Team CSIRO Mineral Resources FlagshipPrivate Bag 10, Clayton South 3169Victoria, Australia

t +61 3 9545 8785 e [email protected] www.csiro.au/cpse

Quantitative Phase Analysis · 2015-08-03 · Quantitative Phase Analysis Theory and Practice Ian C. Madsen CSIRO Mineral Resources Flagship QUANTITATIVE PHASE ANALYSIS WORKSHOP –

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