Quantitative XEDS

• Once we know what our sample consists of, we want to know how much of each element it contains

• We can investigate the relative intensity of characteristic X-ray peaks to get this information

• Current quantification techniques give reasonably accurate results, with room for improvement

Overview

• Fundamental concepts– Castaing– Cliff-Lorimer

• Practical steps– Background substraction– Determine k-factors

• Quantitative analysis

Reading: Williams and Carter, Chapter 35.1-35.8(most important: 35.1-35.4)

Quantitative XEDS: Castaing

• Castaing 1951: the concentration of a present element is proportional to the intensity of the observed characteristic X-ray signal

• Since it is difficult to measure an “absolute” intensity – compare measured value to a standard

where i refers to the specimen value and (i)refers to the standard

Quantitative XEDS: Castaing

K is the sensitivity factor (not constant), determined (inversely) by:

Z the atomic numberA absorption of X-rays within the specimenF fluorescence of X-rays within the specimen

The correction factor for bulk analysis is referred to as ZAF correction

Quantitative XEDS

• HOWEVER…for thin samples A and F are very small and can be ignored– Sensitivity factor proportional only to Z!

• In 1975 Cliff and Lorimer showed that a standard is not needed if intensities for two elements are gathered simultaneously and compared…

Cliff-Lorimer technique

kAB is another sensitivity factor, called the Cliff-Lorimer factor

Ternary systems

Thin-foil criterion

Practical steps for quantification

• Use Kα lines where possible to avoid peak overlap

• Avoid tilt to minimize spurious X-rays• Keep the thinnest part towards the detector• Avoid diffraction conditions (see 35.3)• Collect enough counts for each of the

characteristic peaks – ideally 10000 counts above background for EACH major peak!

Background substraction

• To determine peak intensities you first need to determine the intensity above background

• Bremsstrahlung X-rays decrease continuously in intensity as energy increases– This means that background may not be the same

for each peak, and must be subtracted from each one

Background substraction methods

• Define a ‘window’ spanning the peak – optimum width is 1.2(FWHM), then draw a straight line across/below the peak at this point – everything below is background (35.2)

• Average the bremsstrahlung above and below the peak in the same window (35.3) – note that if there is a significant difference the sample is too thick!

• Model the background mathematically using Kramer’s Law (35.4) – used in commercial software

• Filter out the background mathematically (35.5)

Background substraction methods

Peak integration

• Using a windows method of background estimation: subtract the estimate background from the total intensity in the window

• Kramer’s Law: fit a Gaussian peak and integrate

• Digital filter: match standard peaks stored in a library using least-squares fitting- requires a library of stored spectra; gives a numerical measure of the fit

Determining k-factors

• k-factors can be determined experimentally (using standards) or calculated from first principles

• Remember this is not a constant, but a sensitivity factor that depends on the detector, microscope, analysis conditions…

Determining k-factors - experimental

• We only need to determine kAB in relation to one element. Then all other k-values can be calculated:

• Experimental k-values can be obtained from single-phase compounds

• Standard spectra must be recorded for each instrument setup

Problems determining k-factors

• Existence of appropriate standard – Must be possible to make thin, uniform specimens– No or minimal X-ray absorption from all elements– These thin specimens must be stable under e-beam– Must be stoichiometric throughout the specimen– Si is a useful standard since it exists in many

minerals with other elements, but absorption is very detector-specific

Determining k-factors -theoretical

The accelerating voltage strongly affects QThe atomic number affects ω, a and of course AThe peak integration method also affects aThe detector is of course also a variable (and the window)

Determining k-factors - theoretical

• Experimental k-factors are not really standard (vary with standard quality, analysis conditions, peak integration routines…)

• In many cases a quick answer is required and the highest accuracy is not essential – Calculating k-factors is much easier and faster! And no less

accurate if the standard is uncertain

• Calculations for L-lines are more difficult because of overlap, unknown Q values– No data available for M lines (in most cases)

• Commericial software uses a “black-box” approach: values that are a mix of experimental and calculated, and vary with software package

Zeta-factor method

• Difficulties with Cliff-Lorimer method: determining k, standard samples…

• Zeta-factor method uses pure-element standards – easy to fabricate, not changed by beam

• Major disadvantage: requires in-situmeasurement of probe current hitting specimen: this technology exists but is not yet widely implemented in TEMs

Zeta-factor methods

If the beam current i can be measured and we assume that X-ray intensity is proportional to mass-thickness, ρt (here we neglect absorption and fluorescence):

aiQCNA

CIt

A

AA

ϖζ

ζρ

0

≡

=

where is the zeta factor

Note that N0 is Avogadro’s number. This expression is independent of specimen thickness, composition, and density.

We can also see that:

BB

AA

B

A

II

CC

ζζ

=

Absorption correction

Recall for bulk specimens that the sensitivity factor is a function of:

Z: the efficiency with which an element generates X-rays

Absorption correction, A: X-rays travelling within the solid will have their energy reduced by absorption

Fluorescence, F: High energy X-rays excite lower energy fluorescence radiation

Fluorescence is usually a small effect and will not be considered. However for thick specimens the absorption correction must be considered.

Absorption correction

If one of the elements in a sample absorbs more than another, the generated counts for that element will be lower – so C is not directly proportional to I

We can write

Here kAB is the “true” sensitivity factor (for thickness = 0) and A is the absorption correction factor.

A is a complicated expression, which is integrated over the sample thickness. It is necessary to have information about the specimen density and thickness at each analysis position, including any variations of density with thickness

B

AAB

B

A

IIAk

CC )(=

Example: AuIn nanoparticle

Example: AuIn nanoparticle