quantitative x-ray spectrometry in tem/stem

1PASI - Electron Microscopy - Chile

Lyman - EDS Quant

Quantitative X-ray Spectrometry in TEM/STEM

Charles LymanLehigh UniversityBethlehem, PA

Based on presentations developed for Lehigh University semester courses and for the Lehigh Microscopy School


Lyman - EDS Quant

Quantitative X-ray Analysis of Thin

Specimens Aim of quantitative analysis: to transform the intensities in the X-ray

spectrum into compositional values, with known precision and accuracy

Cliff-Lorimer method:

Precision: collect at least 10,000 counts in the smallest peak to obtain a counting error of less than 3%

Accuracy: measure kAB on a known standard and find a way to handle x-ray absorption effects

How much of each element is present?

€

CA

CB

= kAB

IA

IB

and CA + CB =1

CA = concentration of element AIA = x-ray intensity from element AkAB = Cliff-Lorimer sensitivity factor

What could be simpler?


Lyman - EDS Quant

Assumptions in Cliff-Lorimer Method

Basic assumptions» X-ray intensities for each element are measured simultaneously» Ratio of intensities accounts for thickness variations» Specimen is thin enough that absorption and fluorescence can be ignored

– the “thin-film criterion”– We would like to handle absorption in a better way!

Cliff-Lorimer equation:

» CA and CB are weight fractions or atomic fractions (choose one, be consistent)» kAB depends on the particular TEM/EDS system and kV (use highest kV)

– k-factor is most closely related to the atomic number correction» Can expand to measure ternaries, etc. by measuring more k-factors

€

CA

CB

= kAB

IA

IB

and CA + CB =1


Lyman - EDS Quant

Steps in Quantitative Analysis

Remove background intensity under peaks Integrate counts in peaks Determine k-factors (or -factors) Correct for absorption (if necessary)


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Calculate Background, the Subtract

Gross-Net Method» Draw line at ends of window

covering full width of peak» Impossible with peak overlap» Should work better above 2

keV where background changes slowly

Three-Window Method» Set window with FWHM (or even better

1.2 FWHM)

» Average backgrounds B1 and B2

» Subtract Bave from peak

» Requires well-separated peaks

Background Modeling» Mathematical model of background

as function of Z and E» Useful when peaks are close

together

from Williams and Carter, Transmission Electron Microscopy, Springer, 1996


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Digital Filtering

Convolute spectrum with “top-hat” filter

» Multiply channels of top-hat filter times each spectrum channel

» Place result in central channel

» Step filter over each spectrum channel

Background becomes zero



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Digital Filtering

Spectrum before filteringNote MgK, AlK, and SiK

Spectrum after filteringPositive lobes are

proportional to peak intensities



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Obtaining k-factors

Requirements for standard specimen for k-factor measurement» Single phase (stoichiometric composition helpful)» Homogeneous at the nanometer scale» Thinned to electron transparency without composition change (microtome)» Insensitive to beam damage

Measure k-factors on a known standard:

» Usually kASi or kAFe

» Measure k-factors at various thicknesses and extrapolate to zero thickness

Other ways» Calculate k-factors (when standards are not available)» Use literature values at same kV for x-rays 5-15keV (not recommended)» Use kAB = kAC/kBC (use only when necessary - errors add)

€

kAB =CA

CB

IB

IA


Lyman - EDS Quant

Why Collect 10,000 Counts?

There is a 99% chance that a single measurement is within 3N1/2 of the true value

The relative counting error =

Thus, for 10,000 counts the relative counting error =


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Experimental k-factors



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Calculated k-factors

When suitable standard is not available» When a modestly accurate analysis is acceptable

Most EDS system software can calculate k-factors» But errors can be up to 20%

Simple expression:

€

kAB =Qωa( )B

AA

Qωa( )AAB

εA

εB

but Q not known well which leads to error

Q = ionization cross-section = fluorescence yielda = relative transition probability =A = atomic weight = detector efficiency

€

IntensityKα

Intensity(Kα + Kβ )


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Calculated k-factors

Calculated kAFe-factors using different ionization cross-sections



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kAFe for K-series

Errors of calculated versus standards ~ 4%



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kAFe for L-series

Errors of calcuated versus standards up to 20%



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The Absorption Problem

k-factors measured at different specimen thicknesses will be different» X-rays from some elements will be

absorbed more than others» “Thin-film criterion” breaks down if high

accuracy required» We need a better way to handle

absorption effects

What to do:1. Measure unknown and standard at the

same thickness (impractical)2. Extrapolate all k-factors to zero-

thickness, then apply absorption correction to each measurement (but we need to know the specimen thickness)

3. Use -factors

aa

Incident beamα

XEDStt cosec α



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Extrapolate to the Zero-Thickness k-factor

Zero-thickness k-factor


Horita et al. (1987) and van Cappellan (1990) methods


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Obtaining the Zero-thickness k-factor

Thin standard of known composition

Pt-13wt% Rh thermocouple wire

Thickness measured by EELS log-ratio method

€

kPtRh=CPt

CRh

IRhI Pt

=0.870.13

I RhI Pt

€

kPtRh=1.079

R. E. Lakis, C. E. Lyman, and H. G. Stenger, J. Catal. 154 (1995) 261-275.


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Absorption Correction

Effective sensitivity factor kAB* = kAB(ACF)

€

ACF =

μ

ρ

⎡

⎣ ⎢

⎤

⎦ ⎥Spec

A

μ

ρ

⎡

⎣ ⎢

⎤

⎦ ⎥Spec

B

⎛

⎝

⎜ ⎜ ⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟ ⎟ ⎟

1− exp −μ

ρ

⎡

⎣ ⎢

⎤

⎦ ⎥Spec

B

ρt cosec α( ) ⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥

1− exp −μ

ρ

⎡

⎣ ⎢

⎤

⎦ ⎥Spec

A

ρt cosec α( ) ⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥

⎛

⎝

⎜ ⎜ ⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟ ⎟ ⎟

Equation 35.29:

Zero-thickness k-factor



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Original -factor method

Absorption correction contains t» foil thickness t must be determined at analysis point» specimen density for composition at analysis point)

-factor method» assume x-ray intensity t» then

» subsititute into absorption equation:

€

t = ζ A

IA

CA

€

CA

CB

= kAB

IA

IB

⎛

⎝ ⎜

⎞

⎠ ⎟

μ

ρ

⎡

⎣ ⎢

⎤

⎦ ⎥Spec

A

μ

ρ

⎡

⎣ ⎢

⎤

⎦ ⎥Spec

B

⎛

⎝

⎜ ⎜ ⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟ ⎟ ⎟

1− exp −μ

ρ

⎡

⎣ ⎢

⎤

⎦ ⎥Spec

B

ζ A

IA

IB

⎛

⎝ ⎜

⎞

⎠ ⎟cosec α( )

⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥

1− exp −μ

ρ

⎡

⎣ ⎢

⎤

⎦ ⎥Spec

A

ζ A

IA

IB

⎛

⎝ ⎜

⎞

⎠ ⎟cosec α( )

⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥

⎛

⎝

⎜ ⎜ ⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟ ⎟ ⎟

We can determine both absorption-corrected compositions and t if kAB and known from measurements on standard


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Modified -factor method

Measure the -factor for both elements:

Assume CA + CB = 1 for binary system and rearrange:

Determine CA, CB, and t simultaneously from three equations in three unknowns

t can be determined if density is known

€

t = ζ A

IA

CA

€

t = ζ B

IB

CB

€

CA =ζ AIA

ζ AIA + ζ BIB

, CB =ζ BIB

ζ AIA + ζ BIB

, ρt = ζ AIA + ζ BIB

M. Watanabe and D.B. Williams, Z. Metalkd. 94 (2003) 307-316


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-factor

factor is dependent on • x-ray energy• accelerating voltage• beam current

factor is independent of• specimen thickness• specimen composition• specimen density


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Quantitative analysis by factor method

Lucadamo et al. (1999)


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Effect of kV on Beam Spreading

Elastic scattering broadens the beam as it traverses the specimen

Beam broadening is less for

» Higher kV» Lighter materials» Smaller thicknesses

Goldstein-Reed Eqn.

b=7.21x105 ZE0

1/2

ρA( )

32t

b

bfrom Williams and Carter, Transmission Electron Microscopy, Springer, 1996


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Spatial Resolution vs. Analytical Sensitivity

Conditions that favor high spatial resolution (thinnest specimen) result in poorer analytical sensitivity and vice versa. For example to obtain equivalent analytical sensitivity in an AEM to an EPMA, the X-ray generation and detection efficiency would have to be improved by a factor of 108



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Composition Profiles Across an Interphase Interface

The change in Mo and Cr composition across the interface can be used to determine the compositions of the phases either side of the interface which, in turn, give the tie lines on the Ni-Cr-Mo phase diagram.

Courtesy R. Ayer



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Measurement of Low T Diffusion Data

Measurement of composition profiles with high spatial resolution permits extraction of low- temperature diffusion data because the small diffusion distances at low T are detectable by AEM X-ray microanalysis. Here Zn profiles across a 200 nm wide precipitate-free zone in Al-Zn are used to determine values of the Zn diffusivity at T = 100-200°C. Courtesy A.W. Nicholls

Low-temp data

High-temp data



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Predicted Phase Separation Observed in Nanoparticles

Dotted misibility gap was predicted from other similar systems --> only observed in nanoparticles

Two phases observed

Pt-rich phase

Rh-rich phase

C. E. Lyman, R. E. Lakis, and H. G. Stenger, Ultramicroscopy 58 (1995) 25-34.


Lyman - EDS Quant

Summary

Know the question you are trying to answer

Know the precision and accuracy required to answer the question

Accumulate enough counts in the spectrum to achieve the required precision (> 10,000 counts in the smallest peak)

Know the precision and accuracy of your k-factor

Measure zero-thickness k-factors and apply an absorption correction (need t at analysis point) or use -factors where t is not needed

Spatial resolution vs. detectability: » You cannot achieve the highest spatial resolution and the best analytical sensitivity

under the same experimental conditions

quantitative x-ray spectrometry in tem/stem

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