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 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 ionisation 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 the Photo-electric animation). Clearly, the probability of the X-ray being absorbed is dependent on the other elements in the sample and their ionisation 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).

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.

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 ionisation 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.