2.2. Volume of Excitation

Two factors limit the size and shape of the interaction volume: (1) energy loss through inelastic interactions and (2) electron loss or backscattering through elastic interactions. The resulting excitation volume is a hemispherical to jug-shaped region with the neck of jug at the specimen surface. The analyst must remember that the interaction volume penetrates a significant depth into the sample and avoid edges where it may penetrate overlapping materials. The depth of electron penetration of an electron beam and the volume of sample with which it interacts are a function of its angle of incidence, the magnitude of its current, the accelerating voltage, and the average atomic number (Z) of the sample. Of these, accelerating voltage and density play the largest roles in determining the depth of electron interaction (Figure 2.2a).

Figure 2.2a. Schematic depiction of the variation of interaction volume shape with average sample atomic number (Z) and electron beam accelerating voltage (Eo). The actual shape of the interaction volume is not as long-necked since the electron beam in microprobe analysis has a diameter of about 1 mm (see Figure 2.1b).

Electron penetration generally ranges from 1-5 µm with the beam incident perpendicular to the sample. The depth of electron penetration is approximately (Potts, 1987, p. 336):  

For example, bombarding a material with a density of 2.5 g/cm3, about the minimum density for silicate minerals, with E_o = 15 keV, gives x = 2.3 µm. The width of the excited volume can be approximated by (Potts, 1987, p. 337):

Both of these are empirical expressions. A theoretical expression for the "range" of an electron, the straight line distance between where an electron enters and its final resting place, for a given E_o is (Kanaya & Okayama, 1972):

The volume of interaction can be modeled by Monte Carlo simulation. In such models, the likelihood of incident electrons interacting with the sample and scattering and the angle of deflection are determined probabilistically. X-ray generation depths depend strongly on density and accelerating voltage (Figure 2.2b.). The results derived from Monte Carlo modeling yield a volume of interaction that is very similar to that determined by etching experiments. The excited volume is roughly spherical and truncated by the specimen surface. The depth of the center of the sphere decreases with increasing atomic number of the target.

Figure 2.2b. Comparison of electron paths (top) and sites of X-ray excitation (bottom) in targets of aluminum, copper, and gold at 20 keV, simulated in a Monte Carlo procedure (after Heinrich, 1981).