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.
P2O5
SiO2
TiO2
Al2O3
MgO
CaO
MnO
FeO
Na2O
K2O
Cl
total
o = Cl
total
cations on 10. <o,cl> basis
0.1801
49.6343
1.9337
14.1515
6.9709
11.2084
0.2575
12.2786
2.8663
0.1967
0.0440
99.7219
-0.0099
99.7120
Ratio (Fe+Mn)/(Fe+Mn+Mg) =
P
Si
Ti
Al
Mg
Ca
Mn
Fe
Na
K
Wt.%
0.0786
23.2004
1.1592
7.4896
4.2036
8.0106
0.1994
9.5442
2.1264
0.1633
Cations
0.0093
3.0375
0.0890
1.0207
0.6360
0.7349
0.0133
0.6284
0.3401
0.0154
6.5246
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
Peak
cps
2) background is subtracted
Bkg cps
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 Cstd
Cstd relates concentration in std to pure element
K x 100 = uncorrected wt.%
Corrections and X-Ray Interactions with Matter
Recall that X-rays are generated within the interaction volume
Defined by mean free path of electrons
Critical excitation potential
Incident
beam
Always dealing with measured intensity
of emerging X-rays
Characteristic
X-rays
Sample
Absorption
Fluorescence
Corrections and X-Ray Interactions with Matter
measured
intensity…Ii < I0
Ii
Sample
Absorption
Fluorescence
I0
What affects measured intensity?
Samples and standards are not pure elements = “matrix effects”
1) Differential backscattering
Z
2) Different bulk densities
3) Different scattering and ionization cross sections
4) Differences in the relationship between electron energy loss and
distance traveled (stopping power)
A
5) X-ray absorption
F
6) Secondary fluorescence
Minimize corrections by using standards close in composition and
physical properties to the sample
How do we correct for these effects? Three general
approaches…
ZAF
Generalized algebraic procedure
Generates separate factors for :
Z
atomic number
A
absorption
F
fluorescence
Standard ZAF approach
Φ(ρZ) Use depth distribution of X-ray generation –
express ZAF effects
PAP (Pouchou and Pichoir)
PROZA (Bastin and Heijligers)
X-Phi (Merlet)
Empirical
Based on relative intensities from known specimens
in a specific compositional range
Bence-Albee procedure
One general approach in use today…
Φ(ρZ)
Use depth distribution of X-ray generation –
express ZAF effects
PAP (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
ZAF
Z = 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
Z
Backscattering (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 backscatter
E0 = beam energy
EC = critical excitation potential
Q = ionization cross section
S = electron stopping power
Expression for stopping power (Hans Bethe, 1930)
E = electron energy (eV)
x = path length
e = 2.718 (base of ln)
N0 = Avogadro constant
Z = atomic number
ρ = density
A = atomic mass
J = mean excitation energy (eV)
Or…
Can be expressed as mass distance…
-1/ρ(dE/dx) (in g/cm2)
BSE factor R
Fraction of ionization remaining in target after loss
due to backscattering of beam electrons
Function of atomic # and overvoltage (U)
To evaluate, sum values for all elements present:
For the standard:
From tables
For the sample:
C = wt. fraction of element
R = BSE correction
i = measured element
j = elements present in specimen
Absorption Correction (A)
X-rays absorbed as they pass through
specimen
Reduces 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 Z
Φ(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…
Characteristic
X-rays
Incident
beam
μ / ρ = mass absorption coefficient for the X-ray
Ψ
Ψ = take-off angle
(μ / ρ) cscΨ is referred to as Χ
(chi)
Sample
d is known - solve for λ by changing θ
Move crystal and detector to select different X-ray lines
Crystal
monochromator
Si Kα
S Kα
Cl Kα
Ti Kα
Gd Lα
sample
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 coefficient
Ψ
take-off angle
E0
beam energy
EC
critical excitation potential
Z
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 function
No X-ray production at surface
Philibert
approximation
Φ(ρZ)
True shape
ρZ
What factors increase absorption?
High voltage = deep X-ray production
Low 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)
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 absorption
X-rays from i are fluoresced
So, a sample of olivine has Fe, Mg and Si.
Fe Kα = 6.4 keV
Kα
Binding energies…
Mg K = 1.30 keV
Si K = 1.84 keV
L
K
M
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
*
* = specimen
Ifij = intensity by fluorescence of
element i by element j
Ii = electron generated intensity of i
Sum for all elements
Fluorescence and absorption…
A sample of olivine has Fe, Mg and Si.
Kα
Fe Kα = 6.4 keV
Binding energies…
Mg K = 1.30 keV
Si K = 1.84 keV
L
K
M
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 eV
reduces Mg Kα and Si Kα intensities, so competing factors!
Fluorescence at a distance…
High energy Fe Kα fluoresces Ca Kα in
adjacent phase. Analysis “sees” Ca at
this beam position.
Ca Kα
Ca Kα
Fe-Ca
silicate
Fe
silicate
Fe-Ca
sil.
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 possible
Use standards similar to unknowns in terms of average Z
ZAF correction
1) Determine K for all elements – a first approximation
2) Determine ZAF factors
3) Compute new approximation
4) Compute new ZAF factors
5) Iterate until results converge (usually 2-4 iterations is sufficient)
Important:
Must analyze all elements present in sample
Minimize correction factors by using standards similar to
unknowns
Absorption 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 A
Must still do separate calculation for F…
Φ(ρZ) techniques
For Z
Calculate the area under the Φ(ρZ) curve
For 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
lnΦ(ρZ)
(ρZ)2, mg2/cm4
Means Φ(ρZ) curves are gaussian
centered on the surface of the sample
modify by application of a transient function to make the
curve look like experimental curve
Φ(ρZ)
ρZ
Love-Scott
Use quadrilateral profile and calculate Z and A factors separately
Pouchou and Pichoir (PAP)
Describe Φ(ρZ) curve with pair of intersecting parabolas
Breaks down the curve into four parameters
Calculate the Z correction implicitly on the way to the final formula
Elt.
Peak
Prec.
(Cps)
(%)
Bkgd
(Cps)
Na
K
Mg
Si
Al
P
Cl
Ca
Ti
Mn
Fe
97.0 4.1
16.2 10.1
382.9 1.1
2177.7 0.5
497.7 1.0
2.5 14.1
4.0 11.2
644.6 0.9
104.4 2.2
6.5 8.8
222.3 1.5
3.4
3.9
4.5
12.2
5.2
0.9
2.0
8.6
8.4
2.1
3.7
P/B
28.89
4.18
84.58
178.49
95.08
2.78
1.98
75.39
12.43
3.17
60.92
Ix/ Sig/k
Istd
(%)
0.2071
0.0162
0.3620
0.6044
1.0622
0.0035
0.0006
1.0074
0.0177
0.0057
1.0369
4.2
10.2
1.2
0.5
1.0
14.1
11.3
1.0
2.2
8.8
1.6
Detection Beam
limit (%) (nA)
10.1
0.1141
0.0659
0.0215
0.0461
0.0333
0.0565
0.0339
0.0339
0.0323
0.0680
0.0825
Elt.
Na
K
Mg
Si
Al
P
Cl
Ca
Ti
Mn
Fe
k-ratio
0.0101
0.0014
0.0262
0.1756
0.0521
0.0005
0.0004
0.0725
0.0096
0.0016
0.0788
Correc.
2.1033
1.1396
1.6039
1.3212
1.4378
1.4519
1.2455
1.1054
1.2024
1.2264
1.2110
N = # of counts
99.7% of area
95.4% of area
68.3% of area
3σ
2σ
1σ
1σ
2σ
3σ
Elt.
Na
K
Mg
Si
Al
P
Cl
Ca
Ti
Mn
Fe
O
total :
Conc.
(wt%)
2.1264
0.1633
4.2036
23.2004
7.4896
0.0786
0.0440
8.0106
1.1592
0.1994
9.5442
43.4927
99.7120
1sigma
(wt%)
0.093549
0.024284
0.051090
0.121850
0.089185
0.020254
0.012308
0.083576
0.028746
0.029309
0.171398
Norm Conc.
(wt%)
2.1325
0.1638
4.2158
23.2674
7.5112
0.0788
0.0441
8.0337
1.1626
0.2000
9.5718
43.6183
100.0000
Norm Conc.
(at%)
2.0582
0.0929
3.8486
18.3817
6.1768
0.0565
0.0276
4.4474
0.5385
0.0808
3.8029
60.4882 by stoichiometry
100.0000
Counting statistics here includes both peak and background on both
unknown and calibration standard…
miscellaneo.
P2O5
SiO2
TiO2
Al2O3
MgO
CaO
MnO
FeO
Na2O
K2O
Cl
total
o = Cl
total
cations on 10. <o,cl> basis
0.1801
49.6343
1.9337
14.1515
6.9709
11.2084
0.2575
12.2786
2.8663
0.1967
0.0440
99.7219
-0.0099
99.7120
Ratio (Fe+Mn)/(Fe+Mn+Mg) =
P
Si
Ti
Al
Mg
Ca
Mn
Fe
Na
K
Wt.%
0.0786
23.2004
1.1592
7.4896
4.2036
8.0106
0.1994
9.5442
2.1264
0.1633
Cations
0.0093
3.0375
0.0890
1.0207
0.6360
0.7349
0.0133
0.6284
0.3401
0.0154
6.5246
50.23
All Φ(ρZ) and ZAF corrections depend on the quality of input data
Mass absorption coefficients
Ionization cross sections
Backscatter coefficients
Surface ionization potentials
Because Φ(ρZ) routines model X-ray production near the surface
reasonably well
Can be used on oxides and silicates
Ultralight elements (B, C, N, O)