Quantitative EDS X-ray microanalysis using SEM
How much of each element is present in the sample?
In quantitative EDS microanalysis in SEM, the mass fractions or weight percents of the elements present in the sample are calculated. The spectra are processed to remove Bremsstrahlung X-rays and spectral artifacts, and then the Characteristic X-rays are compared with data measured from standard reference materials. In so called Standardless Quantitative analysis, or semi-quantitative analysis, the spectra are compared with data collected from standards in the factory of the manufacturer of the EDS system and stored with the system software. In fully Standardized Quantitative analysis the spectra from the standards are collected on the same instrument as the spectra from the sample being analysed, which allows for more accurate analyses.
Qualitative analysis to identify the elements present in the sample is a good prelude to quantitative analysis as it will allow the best operating parameters for the microscope to be selected.
Microscope operating parameters
If the sample is stable under high-vacuum in an electron microscope and is not susceptible to damage by the electron beam, then an accelerating voltage of 15-30 kV is recommended for quantitative analysis. This is sufficient to efficiently generate at least one family of X-ray lines for all elements.
If the sample is likely to be damaged by a high-energy primary electron beam, then it may be necessary to use a lower accelerating voltage. In this case, the higher energy X-ray lines may not be efficiently generated and low-energy X-rays will need to be used for element quantification, e.g., the L or M lines for elements with Z > 20. An alternative approach to reducing sample damage is to lower the electron dose by using a broader (defocused) beam.
The electron beam current (probe current or spot size) will control the X-ray count rate or intensity of the generated X-rays. The beam current should be adjusted to minimize damage to the sample but generate sufficient X-rays to allow reliable quantification (see Precision and Accuracy and Detection Limits). At the same time, the beam current needs to be adjusted to minimize spectral artifacts and achieve system dead times of ~20-50%. The beam current needs to be established at the beginning of an analysis session and monitored throughout the session as any drift will affect the number of X-rays generated from the sample which in turn will affect the quantification. Tungsten emitters or hot (Schottky) field emitters are required for quantitative microanalysis. Cold field emitters lack the required beam stability.
The electron beam – sample – detector geometry should be optimized. Set the stage so the sample is at the microscope's manufacturer's recommended working distance. The insertion of the detector is machine-specific; some are fixed and some are movable. In general, the X-ray detector should be as close as possible to the sample to maximize collection of the generated X-rays, but this may be limited by the presence of other detectors and safe operation of the microscope. There must be a clear path between the sample and the X-ray detector.
Sample preparation
For standardized quantitative analysis the samples must be flat and polished. Samples should also be homogeneous, ‘bulk’ not porous or thin films on a substrate, otherwise the matrix correction procedures will not work correctly
Figure: Proper sample preparation is critical for quantitative analysis. If the samples are not (A): polished and void free and (B): homogeneous on the scale of the interaction volume, special analytical techniques must be used.
The size of the particles that can be analysed will depend on their mean atomic number, and the accelerating voltage. (Remember that X-rays will be generated throughout the interaction volume and that the size of the interaction volume depends on kV and Z). Generally, particles to be analysed must be greater than ~2 microns across to be analysed by point analysis.
Ideally, if a conductive coating is required for the sample, the standards should be coated with the same material. Carbon is the recommended coating material as it doesn’t interfere with Characteristic X-ray peaks from elements in the sample.
Limitations of quantitative analysis
Some of the limitations of quantitative EDS analysis are listed below:
- Light elements (Z < 11) cannot be routinely analysed by EDS.
Hydrogen (Z = 1) and He (Z = 2) do not have Characteristic X-rays, and the Li (Z = 3) K X-rays are of too low energy to be detected by EDS.
Beryllium (Z = 4) to Ne (Z = 10) X-rays can be detected by EDS, but there are two problems. Firstly, they are low energy X rays subject to strong absorption by the specimen. Secondly, the electrons involved in generating the Characteristic X-rays are also the valence electrons involved in the chemical bonding of the element, therefore the shapes and positions of the peaks may change in different compounds. The samples and the standards must be closely matched for best results.
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Carbon is the most commonly used coating material for non-conductive samples and cannot be analysed if the sample is carbon coated. A different coating material must be used if analysing for carbon.
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Many minerals contain oxygen bonded with a range of cations. It is common practice to calculate the amount of oxygen in the sample by measuring the percentages of the cations and calculating oxygen by stoichiometry. This generally is more accurate than analysing for oxygen.
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Calculating oxygen by stoichiometry requires knowledge of the valence state of the cations to which it is bonded. This information is not available from EDS analysis
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Some samples may contain structurally bound water or carbonate. Complete analyses of these samples cannot be derived by EDS analysis.
Standardized Quantitative Analysis
Different combinations of electron microscope and ED X-ray detector will involve different protocols for quantitative microanalysis, but there are six basic steps required for Standardized Quantitative analysis that should be common to all systems:
- Define the list of elements that you want to analyse for. Use qualitative analysis to identify all of the elements present in the samples that you want to analyse. Older software packages may only derive quantitative data for a specified element list but newer systems will derive quantitative data for all of the elements detected in the spectrum to be processed.
- Perform any calibration of the EDS system that is necessary for quantitative analysis. At a minimum, it will be necessary to measure the beam current or the X-ray count rate on a pure element standard.
- Measure spectra from standards for the elements that you want to analyse for. Ideally, the spectra will be collected under the same conditions, and on the same instrument, that will be used for collecting spectra from the samples to be analysed. If no standards are available, it may be necessary to use default spectra, and this will affect the accuracy of the analysis.
- Collect spectra from the samples to be analysed. Be sure to monitor the beam current for drift between analyses.
- Process the spectra to calculate the mass fraction or weight percent of the elements present or the percentages of the elements defined in an element list.
- Assess the quality of the analysis. Is it a ‘good’ analysis? If not, why not?
Standardless or semi-quantitative analysis
Most commercial EDS X-ray analysis systems will be installed with default spectra for all elements and the most commonly used X-ray analysis lines. This data allows an estimation of composition to be made for spectra collected on different instruments, and this information can be helpful in identifying different phases in a sample, for example. However, differences between the instrument and conditions for collection of the default spectra and those being used to undertake analysis elsewhere will limit the accuracy and precision of the results, so that they cannot be published as quantitative analyses.
Spectral processing
At the heart of quantitative microanalysis is spectral processing or data reduction. The objective is to calculate the number of X-ray counts in the peak of interest and compare that to the number of X-ray counts in a standard with a known concentration of the element of interest, and from this derive the mass fraction of the element in the sample. There are two steps involved: firstly, the Bremsstrahlung (background or continuum) X-ray counts have to be subtracted from the spectrum, and secondly the Characteristic X-ray peaks have to be processed to derive the concentration of the element concerned.
Different materials have different mean atomic numbers, and because the number of Bremsstrahlung X-rays varies with mean atomic number (Kramer’s Law) the background counts, i.e., the contribution of Bremsstrahlung to the total counts in the X-ray peak, need to be subtracted before the counts in the Characteristic peaks can be calculated. Background subtraction in EDS analysis is most commonly achieved by using a mathematical filter known as a Top-Hat Filter as the first step. The filter is applied to each channel in the spectrum, and a value for the corresponding channel in the filtered spectrum is calculated. The value for any channel in the filtered spectrum can be approximated by finding the average value of the central channel and the three channels on either side of it (2M + 1), and subtracting from that value the average of the four channels on the low-energy side of the ‘hat’ (N), and the average of the four channels on the high-energy sides of the ‘hat’ (N), where M = 3 and N = 4. The result is to remove the slope and noise from the background continuum, and to subtract the background continuum from the Characteristic X ray peaks.
Figure: The application of a top-hat filter to part of an ED spectrum. Optimum values for M and N in the filter are 3 and 4 channels respectively, so the total filter is 15 channels wide.
Concentration calculation
The K ratio is the ratio of the intensity (number of X-ray counts) in the filtered peak for an element of interest in the sample to the intensity in the filtered peak for the standard assigned to that element:
K = Ismpl/Istd
We would expect the concentrations of the element in the sample and the standard to be related to the measured X-ray intensities, so the concentration of the element of interest in the sample can be approximated by the K ratio multiplied by the concentration of the element in the standard, which is known:
Csmpl = K.Cstd
If we apply this relationship to calculate the concentrations of the elements in a sample, we find that some of the calculated concentrations are too low, some are too high and some are about right. It turns out that every element in the sample has an effect on the measured X-ray intensity of every peak in the X-ray spectrum. That is, the measured intensity depends on the composition of the whole sample. Therefore, to calculate the concentrations of the elements in the sample we must apply matrix corrections to the raw intensities to allow for differences in composition between the sample and the standard.
There are three parts to the matrix corrections based on:
- Z - differences in mean atomic number,
- A - differences in absorption of X-rays, and
- F - differences in the production of secondary X-rays, or X-ray fluorescence.
The matrix corrections are therefore commonly known as ZAF corrections.
Atomic number correction, Z
There are two parts to the atomic number correction: a backscattering component and a stopping power component.
Remember that as the mean atomic number of the sample increases the number of electrons that are backscattered (backscatter coefficient) also increases, and this is what gives rise to contrast in backscattered electron images. The electrons that are backscattered are ejected from the sample and cannot generate X-rays from it, so if the sample has a different mean atomic number than the standard a correction to the measured X-ray intensity must be made.
The stopping power is the rate of energy loss by the incident electrons per unit of mass penetrated in the sample, and it decreases with increasing mean atomic number, Z. The mass penetrated increases with increasing Z, and more X-rays are generated from samples with higher Z. The stopping power correction has the opposite sense to the backscatter correction, and the sum of the two corrections makes up the mean atomic number correction, Z.
Absorption correction, A
X-rays generated within the sample travel in all directions through it, and may be absorbed within it. X-rays are either absorbed within the sample or they pass through it – they do not gradually lose energy as electrons do. In the energy range for X-rays generated in the SEM, X-ray absorption is most commonly due to the photo-electric effect. This means that if the energy of the Characteristic X-ray is equal to the ionization energy of an electron shell of an atom in the sample, there is a strong probability that the X-ray photon will be absorbed and a photo-electron will be generated (see Photo-electric animation). Clearly, the probability of the X-ray being absorbed is dependent on the other elements in the sample and their ionization energies.
The probability of the X-ray being absorbed also depends on the distance that it travels through the sample before it escapes and enters the X-ray detector. The path length of the X-ray through the sample is given by z cosec ψ where z is the depth in the sample from which the X-ray is generated and ψ is the takeoff angle of the detector. The absorption correction factor is given by µ cosec ψ where µ is the Mass Absorption Coefficient (MAC).
Figure: The distance travelled through the sample by an X-ray photon generated at depth z is z cosec Ψ, where Ψ is the takeoff angle of the X-ray detector.
In general, MACs increase as the energy of the absorbed X-ray decreases so corrections for low Z elements are large while those for high Z elements are smaller. Also, high Z elements tend to be strong absorbers so large corrections are required for low Z elements in a matrix containing high Z elements.
Mass absorption coefficients
Mass absorption coefficients are stored as a matrix of numbers of absorption of a particular X-ray line (the emitter) by an absorber: For example, a portion of the MAC matrix for Kα X-rays for Z = 23 to 29 is shown below.
| Emitter | V 4952 eV | Cr 5415 eV | Mn 5899 eV | Fe 6403 eV | Co 6930 eV | Ni 7478 eV | Cu 8048 eV |
|---|---|---|---|---|---|---|---|
| V | 94.6 | 73.8 | 498.4 | 403.4 | 328.2 | 268.5 | 220.8 |
| Cr | 111.1 | 86.7 | 68.4 | 454.8 | 370.8 | 303.9 | 250.4 |
| Mn | 125 | 97.6 | 76.9 | 61.3 | 401.9 | 330.1 | 272.4 |
| Fe | 145 | 113.2 | 89.3 | 71.1 | 57.1 | 370.2 | 306 |
| Co | 160.9 | 125.7 | 99.1 | 79 | 63.5 | 51.4 | 329.4 |
| Ni | 187.9 | 146.8 | 115.8 | 92.3 | 74.1 | 60 | 49 |
| Cu | 200.7 | 156.8 | 123.8 | 98.7 | 79.3 | 64.2 | 52.4 |
Note that the MAC for absorption of Fe Kα by Co (79.0) is different from the MAC for absorption of Co Kα by Fe (57.1).
Note also that the MAC for absorption of the Fe Kα X-ray by Cr is high (454.8) while that for absorption of the Fe Kα X-ray by Mn is low (61.3). This means that the Fe Kα X-ray is strongly absorbed by Cr, if it is present in the sample, but not by Mn. In an Fe-Cr alloy the measured intensity of the Fe Kα X-ray will be too low. However, in an Fe-Mn alloy there will be little effect on the measured intensity of the Fe Kα X-ray.
Fluorescence correction, F
The X-rays produced in the sample by the electrons of the primary beam have the potential to produce a second generation of X-rays. This process is known as secondary fluorescence, or just fluorescence. Fluorescence occurs when Characteristic X-rays produced by the primary-beam electrons from one element in the sample have an energy greater than the critical ionization energy of an electron shell in another element present in the sample. For example, Fe Kα X-rays (E = 6.40 keV) are able to fluoresce Cr Kα X-rays (Ec = 5.99 keV), but Cr Kα X-rays (E = 5.41 keV) cannot fluoresce Fe Kα X-rays (Ec = 7.11 keV). In this case, in an Fe-Cr alloy, the measured intensity of the Cr Kα X-ray will be too high although there will be little effect on the measured intensity of the Fe Kα X-ray.
Corrected concentration calculation
The calculation of the concentration of an element in a sample must therefore take account of the differences in composition between the sample and the standard. A ZAF factor can be calculated that takes account of the stopping power, backscattering coefficient, absorption and fluorescence effects.
Csmpl = K.Cstd.ZAFsmpl/ZAFstd
The ZAF factor for the standard can be calculated from its composition, which is known, but the composition of the sample needs to be known before the ZAF factor can be calculated. As the composition of the sample is not known, an iterative technique is used, with an initial composition calculated from the measured K ratio. The ZAF factor for this composition is calculated and the composition of the sample is recalculated, and the process repeated until there is no change in the calculated composition.
