Editing Auger electron spectroscopy (section)

===Quantitative analysis===

Semi-quantitative compositional and element analysis of a sample using AES is dependent on measuring the yield of Auger electrons during a probing event. Electron yield, in turn, depends on several critical parameters such as electron-impact cross-section and fluorescence yield.<ref name="thompson1985"/><ref name="feldman_mayer"/> Since the Auger effect is not the only mechanism available for atomic relaxation, there is a competition between radiative and non-radiative decay processes to be the primary de-excitation pathway. The total transition rate, ω, is a sum of the non-radiative (Auger) and radiative (photon emission) processes. The Auger yield, <math>\omega_A</math>, is thus related to the [[fluorescence]] (x-ray) yield, <math>\omega_X</math>, by the relation,

:<math>\omega_A=1-\omega_X=1-\frac{W_X}{W_X+W_A}</math>

[[File:Auger xray wiki in png format.png|thumb|340px|Figure 4. Fluorescence and Auger electron yields as a function of atomic number for K shell vacancies. Auger transitions (red curve) are more probable for lighter elements, while X-ray yield (dotted blue curve) becomes dominant at higher atomic numbers. Similar plots can be obtained for L and M shell transitions. Coster – Kronig (i.e. intra-shell) transitions are ignored in this analysis.]]

where <math>W_X</math> is the X-ray transition probability and <math>W_A</math> is the Auger transition probability.<ref name="feldman_mayer"/> Attempts to relate the fluorescence and Auger yields to atomic number have resulted in plots similar to figure 4. A clear transition from electron to photon emission is evident in this chart for increasing atomic number. For heavier elements, x-ray yield becomes greater than Auger yield, indicating an increased difficulty in measuring the Auger peaks for large Z-values. Conversely, AES is sensitive to the lighter elements, and unlike [[X-ray fluorescence]], Auger peaks can be detected for elements as light as [[lithium]] (''Z'' = 3). [[Lithium]] represents the lower limit for AES sensitivity since the Auger effect is a "three state" event necessitating at least three electrons. Neither [[hydrogen|H]] nor [[Helium|He]] can be detected with this technique. For K-level based transitions, Auger effects are dominant for ''Z'' < 15 while for L- and M-level transitions, AES data can be measured for ''Z'' ≤ 50.<ref name="feldman_mayer"/> The yield limits effectively prescribe a cutoff for AES sensitivity, but complex techniques can be utilized to identify heavier elements, such as [[uranium]] and [[americium]], using the Auger effect.<ref name="grant2003"/>

Another critical quantity that determines yield of Auger electrons at a detector is the electron impact cross-section. Early approximations (in cm<sup>2</sup>) of the cross-section were based on the work of Worthington and Tomlin,

:<math>\sigma_{ax}(E)=1.3\times10^{13}b\frac{C}{E_p}</math>

with ''b'' acting as a scaling factor between 0.25 and 0.35, and ''C'' a function of the primary electron beam energy, <math>E_p</math>. While this value of <math>\sigma_{ax}</math> is calculated for an isolated atom, a simple modification can be made to account for matrix effects:

:<math>\sigma (E)=\sigma_{ax}[1+r_m(E_p,\alpha)]</math>

where α is the angle to the surface normal of the incident electron beam; ''r<sub>m</sub>'' can be established empirically and encompasses electron interactions with the matrix such as ionization due to backscattered electrons. Thus the total yield can be written as:
:<math>Y(t)=N_x \times \delta t \times \sigma (E,t )[1-\omega_X] \exp\left(-t\cos \frac{\theta}{\lambda}\right) \times I(t)\times T\times\frac{d(\Omega)}{4\pi}</math>

Here ''N<sub>x</sub>'' is the number of ''x'' atoms per volume, λ the electron escape depth, θ the analyzer angle, ''T'' the transmission of the analyzer, ''I(t)'' the electron excitation flux at depth ''t'', dΩ the solid angle, and δt is the thickness of the layer being probed. Encompassed in these terms, especially the Auger yield, which is related to the transition probability, is the quantum mechanical overlap of the initial and final state [[wave function]]s. Precise expressions for the transition probability, based on first-order perturbation [[Hamiltonian (quantum mechanics)|Hamiltonian]]s, can be found in Thompson and Baker.<ref name="thompson1985"/> Often, all of these terms are not known, so most analyses compare measured yields with external standards of known composition. Ratios of the acquired data to standards can eliminate common terms, especially experimental setup characteristics and material parameters, and can be used to determine element composition.<ref name="briggs_sheah1983"/><ref name="feldman_mayer"/><ref name="oura2003"/> Comparison techniques work best for samples of homogeneous binary materials or uniform surface layers, while elemental identification is best obtained from comparison of pure samples.