X-Ray Microanalysis - Determination of Elemental Concentration

How do we get from counts to concentration?

Element Wt.%

Ti 29.9

Fe 35.8

Mn 2.82

O 31.3

total 99.82

miscellaneo. cations on 10. <o,cl> basis

Wt.% Cations P2O5 0.1801 P 0.0786 0.0093 SiO2 49.6343 Si 23.2004 3.0375 TiO2 1.9337 Ti 1.1592 0.0890 Al2O3 14.1515 Al 7.4896 1.0207 MgO 6.9709 Mg 4.2036 0.6360 CaO 11.2084 Ca 8.0106 0.7349 MnO 0.2575 Mn 0.1994 0.0133 FeO 12.2786 Fe 9.5442 0.6284 Na2O 2.8663 Na 2.1264 0.3401 K2O 0.1967 K 0.1633 0.0154 Cl 0.0440 total 99.7219 6.5246 o = Cl -0.0099 total 99.7120

Ratio (Fe+Mn)/(Fe+Mn+Mg) = 50.23

Pulses converted to counts at a selected wavelength or energy corresponding to an element

Intensity (I) = counts per sec / nA

1) counts are corrected for dead time

2) background is subtracted

3) then compare to standard of known composition

For example:

wt.fraction Si = ISiKα (unknown) / ISiKα (pure std.)K-ratio = [ISiKα (unknown) / ISiKα (std.)] x CstdCstd relates concentration in std to pure element

K x 100 = uncorrected wt.%

Peak cps

Bkg cps

Recall that X-rays are generated within the interaction volumeDefined by mean free path of electronsCritical excitation potential

Always dealing with measured intensity of emerging X-rays

Corrections and X-Ray Interactions with Matter

Incident beam

Characteristic X-rays

Sample

AbsorptionFluorescence

measured intensity…Ii < I0

Corrections and X-Ray Interactions with Matter

Sample

AbsorptionFluorescence

I0

Ii

What affects measured intensity?Samples and standards are not pure elements = “matrix effects”1) Differential backscattering2) Different bulk densities3) Different scattering and ionization cross sections4) Differences in the relationship between electron energy loss and

distance traveled (stopping power)5) X-ray absorption6) Secondary fluorescence

Minimize corrections by using standards close in composition and physical properties to the sample

Z

A

F

How do we correct for these effects? Three general approaches…

ZAF Generalized algebraic procedureGenerates separate factors for :

Z atomic numberA absorptionF fluorescence

Standard ZAF approachΦ(ρZ) Use depth distribution of X-ray generation – express ZAF effectsPAP (Pouchou and Pichoir)PROZA (Bastin and Heijligers)X-Phi (Merlet)

EmpiricalBased on relative intensities from known specimens in a specific compositional rangeBence-Albee procedure

One general approach in use today…

Φ(ρZ) Use depth distribution of X-ray generation –

express ZAF effectsPAP (Pouchou and Pichoir)PROZA (Bastin and Heijligers)X-Phi (Merlet)

Many variations in this approach, mainly centering on the estimation of the area constrained by Φ(ρZ)

For any correction procedure to work:1) Sample must be homogeneous in interaction volume –

note – fluorescence range may be quite high - interface problems

2) Must have high polish and must not be tilted relative to the beam

3) No use of chemical etching or polishing techniques

ZAFZ = atomic number factor (matrix effects and beam electrons)Backscattering (R)Electron stopping power (S)

Expression for average Z…

Low ρ and ave Z

High ρ and ave Z

ZBackscattering (R)Electron stopping power (S)

Duncomb and Reed (1968)

Ri = BSE correction factor for element i in sample (*) and standard

= photons generated / photons generated without backscatterE0 = beam energyEC = critical excitation potentialQ = ionization cross sectionS = electron stopping power

Or…

E = electron energy (eV)x = path lengthe = 2.718 (base of ln)N0 = Avogadro constantZ = atomic numberρ = densityA = atomic massJ = mean excitation energy (eV)

Expression for stopping power (Hans Bethe, 1930)

Can be expressed as mass distance… -1/ρ(dE/dx) (in g/cm2)

BSE factor RFraction of ionization remaining in target after loss due to backscattering of beam electronsFunction of atomic # and overvoltage (U)To evaluate, sum values for all elements present:

For the standard:

For the sample:

C = wt. fraction of elementR = BSE correctioni = measured elementj = elements present in specimen

From tables

Absorption Correction (A)X-rays absorbed as they pass through specimenReduces the observed intensity, following a Beer-Lambert relationship

Sheffield Hallam Chemistry

Castaing (1951)Intensity of characteristic radiation (no absorption case)

Intensity of element i from layer of thickness dZ of density ρ at depth ZΦ(ρZ) is the distribution of characteristic X-ray production with depth

The total flux for element I (no absorption), is then…

And the total flux with absorption is then…

μ / ρ = mass absorption coefficient for the X-ray Ψ = take-off angle(μ / ρ) cscΨ is referred to as Χ (chi)

Incident beamCharacteristic

X-rays

SampleΨ

d is known - solve for λ by changing θMove crystal and detector to select different X-ray lines

Si Kα

S Kα

Cl Kα

Ti Kα

Gd Lαsample

Crystal monochromator

Proportional counter Maintain Bragg condition = motion of

crystal and detector along circumference of circle (Rowland circle)

If generated intensity is F(0) when X = 0 and emitted intensity is F(X)Then we can define F(X) / F(0) as f (X)Which is formulated as…

And the absorption correction is…

The absorption correction factor f(X) for a characteristic X-ray of element i is a function of:μ / ρ mass absorption coefficientΨ take-off angleE0 beam energyEC critical excitation potentialZ atomic #A atomic wt.

Therefore…

The calculation of f(x) includes the estimation of Φ(ρZ), which can be done in a number of ways

The approach in standard ZAF uses the Philibert approximation, which treats Φ(ρZ) as an exponential functionNo X-ray production at surface

Φ(ρZ)

ρZ

Philibert approximation

True shape

What factors increase absorption?High voltage = deep X-ray productionLow take-off angle High μ / ρ

like soft X-rays in matrix with heavy atoms

Functionality of Philibert expression for Φ(ρZ) breaks down in high absorption situations and leads to large errors

Standard ZAF is good for metals

Not good for oxides, silicates

Poor for ultralight elements (CNO)

Kα

KL

M

Fluorescence factor (F)If the energy of a characteristic X-ray from element j exceeds the critical excitation potential for element i, can get photoelectric absorptionX-rays from i are fluoresced

So, a sample of olivine has Fe, Mg and Si.

Fe Kα = 6.4 keV

Binding energies…Mg K = 1.30 keVSi K = 1.84 keV

So Fe Ka excites both Si Kα and Mg Kα, resulting in “too much” intensity for Mg and Si

Fluorescence factor (F)Electrons attenuated more effectively than photons, so fluorescence range can be considerably larger than interaction volume

*

* = specimenIfij = intensity by fluorescence of element i by element jIi = electron generated intensity of iSum for all elements

Kα

KL

M

Fluorescence and absorption…

A sample of olivine has Fe, Mg and Si.

Fe Kα = 6.4 keV

Binding energies…Mg K = 1.30 keVSi K = 1.84 keV

Fe Kα excites both Si Kα and Mg Kα, resulting in “too much” intensity for Mg and Si, meanwhile, the Fe Kα intensity decreases due to absorption by Si and Mg, resulting in “too little” intensity of Fe…

However, Fe LIII edge (binding energy) = 707 eVreduces Mg Kα and Si Kα intensities, so competing factors!

Fe-Ca silicate

Fe-Ca sil.

Ca KαCa Kα

Fluorescence at a distance…

Fe silicate

High energy Fe Kα fluoresces Ca Kα in adjacent phase. Analysis “sees” Ca at this beam position.

In many cases, must correct for fluorescence caused by background radiation

Very important when analyzing a minor amount of a heavy element in a light matrix (Ti in quartz!)

For this reason, if looking for trace elements in light matrix:Choose the softest (lowest energy) line possibleUse standards similar to unknowns in terms of average Z

ZAF correction 1) Determine K for all elements – a first approximation2) Determine ZAF factors3) Compute new approximation4) Compute new ZAF factors5) Iterate until results converge (usually 2-4 iterations is sufficient)

Important:Must analyze all elements present in sampleMinimize correction factors by using standards similar to unknownsAbsorption corrections can be quite substantial in silicates and oxides, so standard ZAF not used for these materials

Use: Φ(ρZ) or Bence-Albee (empirical)

Φ(ρZ) techniques

Obtain f(X) by using equations that describe Φ(ρZ) curves for various elements, X-ray lines, and beam voltages The object, therefore, is to develop a mathematical expression designed to match experimental curves, e.g.Φ(ρz) = γ exp - α2(ρz)2 { 1- [(γ - Φ(0)) exp - βρz ] / γ } (Packwood and Brown)Can then determine corrections for Z and AMust still do separate calculation for F…

Φ(ρZ) techniques

For Z Calculate the area under the Φ(ρZ) curveFor A Express f(X) in terms of Φ(ρZ) The combined expression is then…[ γ R(X /2α) - (γ - Φ(0)) R(( β + X) / 2 α)] / α-1

ZiAi = [γ R(X /2 α) - (γ - Φ (0)) R((β + X) / 2 α)]*/ α *-1

Can then get the complete ZAF correction by combining with standard F expression

How to determine Φ(ρZ) curves – different models

Packwood and Brown (1981)Plot Φ(ρZ) vs. (ρZ)2 = straight line beyond Φ max

(ρZ)2, mg2/cm4

lnΦ(ρZ)

Means Φ(ρZ) curves are gaussiancentered on the surface of the samplemodify by application of a transient function to make the curve look like experimental curve

Φ(ρZ)

ρZ

Love-ScottUse quadrilateral profile and calculate Z and A factors separately

Pouchou and Pichoir (PAP)Describe Φ(ρZ) curve with pair of intersecting parabolasBreaks down the curve into four parametersCalculate the Z correction implicitly on the way to the final formula

Elt. Peak Prec. Bkgd P/B Ix/ Sig/k Detection Beam (Cps) (%) (Cps) Istd (%) limit (%) (nA) 10.1 Na 97.0 4.1 3.4 28.89 0.2071 4.2 0.1141 K 16.2 10.1 3.9 4.18 0.0162 10.2 0.0659 Mg 382.9 1.1 4.5 84.58 0.3620 1.2 0.0215 Si 2177.7 0.5 12.2 178.49 0.6044 0.5 0.0461 Al 497.7 1.0 5.2 95.08 1.0622 1.0 0.0333 P 2.5 14.1 0.9 2.78 0.0035 14.1 0.0565 Cl 4.0 11.2 2.0 1.98 0.0006 11.3 0.0339 Ca 644.6 0.9 8.6 75.39 1.0074 1.0 0.0339 Ti 104.4 2.2 8.4 12.43 0.0177 2.2 0.0323 Mn 6.5 8.8 2.1 3.17 0.0057 8.8 0.0680 Fe 222.3 1.5 3.7 60.92 1.0369 1.6 0.0825

Elt. k-ratio Correc. Na 0.0101 2.1033 K 0.0014 1.1396 Mg 0.0262 1.6039 Si 0.1756 1.3212 Al 0.0521 1.4378 P 0.0005 1.4519 Cl 0.0004 1.2455 Ca 0.0725 1.1054 Ti 0.0096 1.2024 Mn 0.0016 1.2264 Fe 0.0788 1.2110

68.3% of area95.4% of area99.7% of area

3σ 2σ 1σ 1σ 2σ 3σ

 

N = # of counts

     

Elt. Conc. 1sigma Norm Conc. Norm Conc. (wt%) (wt%) (wt%) (at%) Na 2.1264 0.093549 2.1325 2.0582 K 0.1633 0.024284 0.1638 0.0929 Mg 4.2036 0.051090 4.2158 3.8486 Si 23.2004 0.121850 23.2674 18.3817 Al 7.4896 0.089185 7.5112 6.1768 P 0.0786 0.020254 0.0788 0.0565 Cl 0.0440 0.012308 0.0441 0.0276 Ca 8.0106 0.083576 8.0337 4.4474 Ti 1.1592 0.028746 1.1626 0.5385 Mn 0.1994 0.029309 0.2000 0.0808 Fe 9.5442 0.171398 9.5718 3.8029 O 43.4927 43.6183 60.4882 by stoichiometry

total : 99.7120 100.0000 100.0000

Counting statistics here includes both peak and background on both unknown and calibration standard…

miscellaneo. cations on 10. <o,cl> basis

Wt.% Cations P2O5 0.1801 P 0.0786 0.0093 SiO2 49.6343 Si 23.2004 3.0375 TiO2 1.9337 Ti 1.1592 0.0890 Al2O3 14.1515 Al 7.4896 1.0207 MgO 6.9709 Mg 4.2036 0.6360 CaO 11.2084 Ca 8.0106 0.7349 MnO 0.2575 Mn 0.1994 0.0133 FeO 12.2786 Fe 9.5442 0.6284 Na2O 2.8663 Na 2.1264 0.3401 K2O 0.1967 K 0.1633 0.0154 Cl 0.0440 total 99.7219 6.5246 o = Cl -0.0099 total 99.7120

Ratio (Fe+Mn)/(Fe+Mn+Mg) = 50.23

All Φ(ρZ) and ZAF corrections depend on the quality of input dataMass absorption coefficientsIonization cross sectionsBackscatter coefficientsSurface ionization potentials

Because Φ(ρZ) routines model X-ray production near the surface reasonably well

Can be used on oxides and silicatesUltralight elements (B, C, N, O)