5.4.2. Absorption Correction (A)
Since X-rays are generated below the surface of the sample, the emergent radiation suffers absorption prior to detection. The absorption correction is a function of the take-off angle (length of path traversed by the X-rays), the distribution of X-ray generation, the wavelength of the emergent X-ray and the elements present (Figure 5.4.2a). Several models have been used to describe the depth distribution of X-ray generation termed "f(rZ)" (Figure 5.4.2b). As the take-off angle increases, the intensity of characteristic radiation decreases due to an increase in path length. Less energetic X-rays are more easily absorbed. Absorption can also be strongly affected by surface irregularities -- a good sample polish is thus critical.
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Figure 5.4.2a. The forms of the depth and lateral generated intensity functions f(rZ) and y(y). The vertical dimension in this figure is depth beneath the sample surface, expressed as mass thickness, rz, and the horizontal dimension is lateral distance from the electron beam axis, expressed in arbitrary units, y. Most X-rays are generated at relatively shallow depths within the excitation volume and relatively close to the beam axis; this is the region in which electron energies have not been greatly attenuated by ionization or electron scattering (after Williams 1987). |
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Figure 5.4.2b. The Bishop rectangular approximation of f(r,Z), compared to experimental data and Philibert's analytical approximation. The area of the Bishop rectangle is assumed to approximate to that under the f(r,Z) curve, and the continuous variation in f(r,Z) is replaced by the concept of a 'mean mass depth' at half the depth of the Bishop rectangle. This greatly simplifies the calculations required to correct for the effects of variable compositions on the depth generation of analytical X-rays (after Williams 1987). |
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As noted above, absorption can be described
by Lambert's Law. The absorption factor for samples examined by microprobe
analysis, Fa, is expressed by the Philibert-Duncumb-Heinrick
equation. This equation accounts for the effects of take-off angle, accelerating
voltage, composition, etc.:
where,
Recall, Z= atomic number, A = atomic weight, m = mass absorption coefficient, f = take-off angle, Eo = accelerating voltage, and Ec = critical ionization potential. For compounds the mean value of h is used:
where, ci = mass concentration of element i.
The sigma factor above accounts for the voltage dependence of absorption of primary electrons. Errors in the calculation of Fa can be decreased by using a high take-off angle to minimize c and low overvoltages to maximize s.