Auger effect

​⟨f∣H^int​∣i⟩

​2ρ(E), where $|i\rangle$∣i⟩ represents the initial core-hole state (e.g., a single vacancy in the K-shell with filled valence shells), $|f\rangle$∣f⟩ is the final state consisting of a continuum Auger electron and a two-hole dicationic ion, $\hat{H}_\text{int} = \sum_{j<k} \frac{e^2}{r_{jk}}$H^int​=∑j<k​rjk​e2​ is the Coulomb repulsion Hamiltonian between electrons, and $\rho(E)$ρ(E) is the density of final states at the excess energy $E = E_i - E_f$E=Ei​−Ef​. This expression captures the second-order process where a valence electron fills the core vacancy, transferring energy to eject another valence electron. The matrix element $\langle f | \hat{H}_\text{int} | i \rangle$⟨f∣H^int​∣i⟩ involves integrals over the overlapping radial wave functions of the core, valence, and continuum electrons, which determine the transition strength.^[20] For a specific K-LM Auger transition, the kinetic energy of the ejected electron is given by the energy balance: $$E_\text{kin} = E_K - E_L - E_M - U,$$Ekin​=EK​−EL​−EM​−U, where $E_K$EK​, $E_L$EL​, and $E_M$EM​ are the binding energies of the K-, L-, and M-shell electrons, respectively, and $U$U accounts for the binding energy of the final dicationic state (including electron-electron repulsion and relaxation effects). This formula highlights how the available excess energy dictates the continuum state's momentum and influences the density of states $\rho(E)$ρ(E) in the rate expression, with higher energies typically leading to broader spectral features. In atomic systems, $U$U is often small compared to shell bindings but becomes significant in solids due to extra-atomic relaxation.^[21] The dependence of the Auger rate on atomic number $Z$Z arises primarily from the scaling of wave function overlaps and continuum normalization in the matrix element. For high $Z$Z, the rate scales approximately as $Z^4$Z4 due to the increased localization of inner-shell orbitals, enhancing the Coulomb interaction strength; however, for low $Z$Z, screening by outer electrons weakens this scaling, resulting in lower effective rates. The lifetime of the core-hole excited state is inversely related to the total rate via $\tau = 1/A$τ=1/A, typically yielding ultrafast decays. Computational evaluation of these rates employs self-consistent field methods, such as Hartree-Fock (including relativistic Dirac-Hartree-Fock variants) or density functional theory, to generate accurate single-particle orbitals for the matrix elements and density of states; these approaches account for configuration interaction to improve precision for multi-electron systems. Typical rates for core-hole decays fall in the range $10^{14}$1014 to $10^{16}$1016 s$^{-1}$−1, corresponding to lifetimes of 10 to 100 femtoseconds.^[22] As a representative example, the total K-shell Auger rate for neutral neon ($Z = 10$Z=10) is approximately $1.9 \times 10^{15}$1.9×1015 s$^{-1}$−1, dominated by transitions to $\text{Ne}^{2+}$Ne2+ final states like $1s^2 2s^2 2p^4$1s22s22p4 and $1s^2 2s 2p^5$1s22s2p5; this yields a core-hole lifetime of about 500 attoseconds, underscoring the femtosecond-scale dynamics of the process. Such rates have been benchmarked against experimental linewidths in photoelectron spectroscopy, validating the theoretical framework for light elements.^[23]

Selection Rules and Parameters

Experimental Observation

Detection Methods

Spectroscopy Techniques

Applications

Surface and Material Analysis

Medical and Biological Uses