Practical Electron Microscopy and Database

In Electron Energy Loss Spectroscopy (EELS), the quantification of elemental species is fundamentally based on the relationship between the element-specific core-loss signal and the areal density of atoms. The areal density N_k​ of a given element k​ is directly proportional to the core-loss signal I_k​, given by:

-------------------------------------------- [4769a]

where,
           I_t​ represents the total spectrum intensity.
           σ_k denotes the core-loss edge cross-section for inelastic scattering.

A proper approximation can be made simply by use of a low-loss normalization function, given by, [1]

----------------------------------------- [4769b]

where,
           I_k​(β, Δ) is the intensity of the core-loss signal for element k, integrated over the effective collection angle β and a specific energy loss window Δ.
           I_low(β, Δ) is the intensity of the low-loss region, integrated over the same angle β and energy window Δ.
           σ_k(β, Δ) is the cross-section, modified to account for the specific angle and energy window.           

The integration over an effective capture angle β and an energy loss window Δ takes into account the practical considerations of how EELS data is typically collected. This is important because the signal I_k​ and the corresponding cross-section σ_k​ are dependent on both the angle of collection and the energy window.

The core-loss intensity integrated over both the angular range up to β and the energy window Δ, I_k​(β, Δ), can be given by,

----------------------------------------- [4769c]

where,
            is the double differential cross-section, which gives the intensity of the core-loss signal as a function of scattering angle θ and energy loss E. The double differential cross-section, determining the scattering intensity, depends on energy loss and scattering angles is discussed on page3. The cross-section provides the probability of an electron being scattered into a particular angle θ with a specific energy loss ΔE.
           θ represents the scattering angle, and the integration over θ is up to the effective capture angle β.
           E represents the energy loss, and the integration over E is within the chosen energy window Δ.
             represents the integration of the differential intensity over the angular range from 0 to β. This captures the contribution of all scattered electrons up to the angle β.
             represents the integration over the specific energy loss range Δ. This captures the portion of the core-loss signal within that energy window. Choosing different Δ values changes the part of the spectrum being integrated, which can affect the observed intensity depending on how much of the core-loss edge and fine structure are included.

Figure 4769a shows the dependence of 3D surface plot of I_k​(β, Δ) on effective capture angle in radians, β, and energy window Δ in eV.

As β increases, more of the scattered electrons are included in the measurement, which typically increases the intensity I_k​ but also introduces more plural scattering contributions.

If we make certain assumptions, such as a small capture angle where the angular distribution is approximately constant, or a small energy window where the energy dependence can be approximated as constant, we might simplify Equation 4769c to:

----------------------------------------- [4769d]

where,
           E₀​ is a representative energy loss within Δ.
            is the intensity per unit angle and energy at the center of the angular and energy window.

This simplified form gives an intuitive understanding that I_k​(β, Δ) scales with the capture angle β\beta β and the energy window Δ, but in practice, the full double integration is necessary to account for the actual distribution of intensity in both angle and energy.

In EELS intensity can be used to evaluate the areal density of atoms in nanomaterials. [2] In simplified practice, scattering probability can be calculated by:

----------------------------------------------------------- [4769e]

where,

I(ΔE) is the intensity due to inelastically scattered electrons.

I₀​ is the incident beam intensity.

Equation 4769e gives the probability P that an electron will undergo inelastic scattering and contribute to the EELS signal. The areal density N can be obtained by integrating the intensity of the EELS signal (after background subtraction) over the core-loss edge:

----------------------------------------------------------- [4769e]

where,

I_edge​ is the integrated intensity of the EELS signal over the core-loss edge.

σ_atom​ is the partial cross-section for inelastic scattering for a single atom, which can be obtained using theoretical models like the hydrogenic approximation.

This approach allows to accurately measure the areal density of atoms in very thin samples, overcoming some of the limitations of traditional thickness measurement methods like the log-ratio method as shown in Figure 4769b. The method is particularly effective for materials where accurate thickness measurement is challenging due to factors like surface contamination or complex composition. This approach involves correlating the measured EELS signal, which arises from inelastic scattering of electrons, with the number of atoms per unit area in the sample. By doing so, they were able to obtain quantitative information about the areal density of specific elements within nanomaterials. This method is particularly useful in the characterization of materials where atomic-scale resolution is required to understand material properties and behaviors.

[1] R.F. Egerton, Ultramicroscopy 3 (1978) 243–251.
[2] Mengkun Tian, Ondrej Dyck, Jingxuan Ge, and Gerd Duscher, Measuring the areal density of nanomaterials by electron energy-loss spectroscopy, 196, (2019), 154-160.