Physical Principles of Electron Microscopy: An Introduction to TEM ...

158 Chapter 6
Many-electron atoms represent a difficult theoretical problem because, in
a classical (particle) model, the distance between the different orbiting
electrons is always changing. In any event, a more realistic conception of the
atom uses wave mechanics, treating the atomic electrons as de Broglie
waves. Analysis then involves solving the Schrödinger wave equation to
determine the electron wavefunctions, represented by orbitals (pictured as
charge-density clouds) that replace the concept of particle orbits. An exact
solution is possible for hydrogen and results in binding energies that are
identical to those predicted by Eq. (6.4). Approximate methods are used for
the other elements, and in many cases the calculated energy levels are
reasonably close to those determined from optical spectroscopy.
Other wave-mechanical principles determine the maximum number of
electrons in each atomic shell: 2 for the innermost K-shell, 8 for the L-shell,
18 for the M-shell and so forth. Because an atom in its ground state
represents the minimum-energy configuration, electrons fill these shells in
sequence (with increasing atomic number), starting with the K-shell.
As indicated by Table 6-1, the measured energy levels differ substantially
between different elements, resulting in photon energies (always a difference
in energy between two atomic shells) that can be used to identify each
element. Except for H, He, and Li, these photon energies are above 100 eV
and lie within the x-ray region of the electromagnetic spectrum.
6.2 X-ray Emission Spectroscopy
When a primary electron enters a TEM or SEM specimen, it has a (small)
probability of being scattered inelastically by an inner-shell (e.g. K-shell)
electron, causing the latter to undergo a transition to a higher-energy orbit
(or wave-mechanical state) and leaving the atom with an electron vacancy
(hole) in its inner shell. However, the scattering atom remains in this excited
state for only a very brief period of time: within about 10 -15 s, one of the
other atomic electrons fills the inner-shell vacancy by making a downward
transition from a higher energy level, as in Fig. 6-1b. In this de-excitation
process, energy can be released in the form of a photon whose energy (hf ) is
given roughly by Eq. (6.5) but more accurately by the actual difference in
binding energy between the upper and lower levels.
The energy of this characteristic x-ray photon therefore depends on the
atomic number Z of the atom involved and on the quantum numbers (nl , nu)
of the energy levels involved in the electron transition. Characteristic x-rays
can be classified according to the following historical scheme. The electron
shell in which the original inner-shell vacancy was created, which
corresponds to the quantum number nl , is represented by an upper-case