Practical Electron Microscopy and Database

Much of the spectral information obtainable from EELS is similar to that given by synchrotron-XAS (x-ray absorption spectroscopy), so that EELS in TEM has been referred to as a synchrotron in electron microscopes [1]. The electric field of an incident high energy x-ray can eject an electron of condensed matter. Page4777 lists the comparison between XAS and EELS. One important difference from EELS is that x-rays transfer all their energy by ejecting electrons from their initial state in the material.

Irradiating by the x-ray, the electrons in the sample feel a force parallel to the polarization, resulting in a change of momentum. This force is due to a coupling between the electron and the x-ray (of angular frequency ω), given by,

        ----------------------------- [4778a]

where,
          ε -- Unit vector specifying the polarization of the x-ray;
          d -- Dipole operator, given by;
          E₀ -- Magnitude of the x-ray electric field.

        ----------------------------- [4778b]

where,
          -e -- Charge of the electron.

The cross-section for x-ray absorption from the ground state |ψ₀> of the Hamiltonian H₀ by applying Fermi's golden rule can be given by, [2]

        ----------------------- [4778c]

or,

        ----------------------- [4778d]
        ----------------------- [4778e]

where,
          ħ -- Reduced Planck constant;
          c -- Speed of light.

The most common way of performing XAS is to let x-rays penetrate a thin foil with typical thickness of 10 - 100 µm and then measure the fraction of the incident beam intensity penetrating the foil as a function of the energy of the incident x-rays. Therefore, an x-ray source, such as synchrotrons, that is intense over a large energy range is needed. For energies below an absorption edge, the x-rays penetrate rather easily without absorption because the Pauli exclusion principle prevents excitation, while above the absorption edge threshold the x-rays have sufficient energy to eject core electrons to empty states above the Fermi level.

X-ray absorption extended fine structure (EXAFS) analysis generally uses inner-shell edges with binding energies of order 10 kV that provide a range of 1000 - 2000 eV of fine structure information. EXAFS is particularly useful for amorphous and highly disordered materials.

The first use of density functional theory (DFT) for the calculation of X-ray absorption spectra was done by Müller et al. using a linearized augmented plane waves method in the late 70s [3].

[1] Brown L M 1997 A synchrotron in a microscope Proc. EMAG97 (Cambridge) (Inst. Phys. Conf. Ser. 153) pp 17–21
[2] Fermi, E., Nuclear Physics. University of Chicago Press, 1950.
[3] Müller, J.E., Jepsen, O., 1978. Systematic structure in the K-edge photoabsorption spectra of the 4d transition metals: theory. Phys. Rev. Lett. 40 (11), 720 - 722.