# Quantitative X-Ray Microanalysis

## k-ratio

There are several questions that come up during electron beam studies. The first question is: "What does my specimen look like?" Next comes: "What elements are present in that spot/region?" Finally, the question becomes: "How much of this element is in the spot/region?" Quantitative X-ray microanalysis provides the answer to this last question. In this analysis, we are concerned with the concentrations of elements present in the excited volume of the specimen. Generally, the process involves gathering the net peak intensity (subtracting the background) from the specimen’s spectrum and then dividing that result by the net intensity for the X-ray lines from the standard. These measurements must be made under the same conditions. This resulting equation is known as the k-ratio.

Equation (1), k-ratio: Cspec/Cstd = Ispec/Istd = kspec

The k-ratio is the first approximation of the concentration of a particular element in the specimen. Cspec and Cstd represent the weight percent (mass) concentration of a given element in the unknown (Cspec) and the standard (Cstd). We know the value of Cstd and our goal is the knowledge of Cspec. X-ray intensity measurements used in the k value equation (Ispec/Istd), have been background corrected and any peak overlaps have been addressed.

Inter-element effects occur within the excited volume of the specimen during analysis. These effects modify or correct the k ratio to compensate for these effects. Before an accurate analysis can be performed, one must understand and compensate for these effects, called matrix effects. These are the atomic number effect (Z), the absorption effect (A), and the fluorescence effect (F); known collectively as ZAF.

Equation (2): Cspec/Cstd = [ZAF]spec* Ispec/Istd = [ZAF]spec*kspec

## ZAF

The atomic number effect (Z) is related to the depth to which a beam electron penetrates into the specimen. The Z factor determines the amount of X-ray intensity generated from a specimen. It can be calculated by determining the X-ray generation in depth as a function of atomic number and electron beam energy.

Z is the product of two factors: backscattering (R) and stopping power (S). The stopping power factor (S) expresses the loss of energy beam electrons experienced within the excited volume of the specimen. Since the loss decreases with increasing atomic number, more X-rays are produced in higher Z targets. Backscattering (R) increases with atomic number. In high Z targets, beam electrons fail to deeply penetrate the specimen and so fewer X-rays are produced. In most cases R and S cancel each other out. Dimensions of X-Ray Interaction Volume

The X-ray absorption effect (Ai) occurs when inner shell ionizations give rise to characteristic X-rays that are created over a range of depths below the surface of the specimen. This happens when a beam electron induced X-ray "A" is completely absorbed by a matrix atom "B" of higher atomic number. The result of this effect is a suppression of the X-ray intensity of element "A", and leads to an underrepresentation of element "A" in the analytical total. The relationships between the initiating atom "A" (referred to as the emitter) and the absorbing atom "B" are documented in tables known as Mass Absorption Coefficients (MAC). These tables permit estimation of the effects of an element on X-rays from another element, which are used in quantitative calculations. The correction is based on the measured X-ray line, sample matrix (or constituent elements), electron beam energy, and the X-ray detection geometry.

The X-ray fluorescence effect (F) This occurs when characteristic X-rays are emitted as a result of photoelectric absorption ionizing inner shell electrons. F occurs when characteristic X-rays are emitted as a result of photoelectric absorption ionizing inner shell electrons. F is the least important factor in the calculation of composition. In fluorescence, an energetic X-ray from element "A" in the specimen matrix interacts with atom "B" in the matrix. The result is that an X-ray from element "B" is generated and produces an erroneous increase in the measured intensity of element "B", leading to an overrepresentation of element "B" in the total. 