4.3: Back to Basics
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As energy of the x-ray is increased, progressively more core electrons are excited and a more atomic picture can be constructed. Terminology:
Sub-shells (connected to the angular Momentum Quantum Number): An atom's electron shells are filled according to the following theoretical constraints:
- Each s subshell holds at most 2 electrons
- Each p subshell holds at most 6 electrons
- Each d subshell holds at most 10 electrons
- Each f subshell holds at most 14 electrons
- Each g subshell holds at most 18 electrons
Shells (principal quantum number): Electron shells are labeled K, L, M, N, O, P, and Q; or 1, 2, 3, 4, 5, 6, and 7; going from innermost shell outward:
Shells | s | p | d | f | g | Total |
---|---|---|---|---|---|---|
K | 2 | 2 | ||||
L | 2 | 6 | 8 | |||
M | 2 | 6 | 10 | 18 | ||
N | 2 | 6 | 10 | 14 | 32 | |
O | 2 | 6 | 10 | 14 | 18 | 50 |
As you tune the applied x-ray photons, resonances with different ionization thresholds for electrons will occur resulting in progressively higher energy (higher energy = higher binding energy) electrons being ejected. This is a signature of the nature of the element.
Orbital energy levels
Assume there is one electron in a given atomic orbital in a hydrogen-like atom (ion). The energy of its state is mainly determined by the electrostatic interaction of the (negative) electron with the (positive) nucleus. The energy levels of an electron around a nucleus are given by :
\[E_{n}=-h c R_{\infty} \frac{Z^{2}}{n^{2}} \nonumber \]
(typically between 1 eV and 103 eV), where \(R_{\infty}\) is the Rydberg constant, \(Z\) is the Atomic number, \(n\) is the principal quantum number, \(h\) is Planck's constant, and c is the speed of light. For hydrogen-like atoms (ions) only, the Rydberg levels depend only on the principal quantum number \(n\).
For multi-electron atoms, interactions between electrons cause the preceding equation to be no longer accurate as stated simply with \(Z\) as the atomic number. Instead an approximate correction may be used where \(Z\) is substituted with an effective nuclear charge symbolized as \(Z_{eff}\).
\[E_{n, l}=-h c R_{\infty} \frac{Z_{e f f^{2}}}{n^{2}} \label{multi} \]
Hence, different atoms (higher \(Z\)) will have different effective \(Z_{eff}\) that couple into Equation \ref{multi} to result in different energies for the electrons. Note chance in energies for the S electrons below:
Hence, XPS can be used for elemental analysis of unknown compounds, by measuring and check to known values for the electronic energy levels of elements