Practical Electron Microscopy and Database

The intensity of the core-loss signal as a function of scattering angle θ and energy loss E in Electron Energy Loss Spectroscopy (EELS) is described by the differential cross-section:

  -------------------------------------------------------- [3a]

where,
           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.

The differential cross-section in Formula 3a depends on several factors including the material properties, the electron energy, and the nature of the inelastic scattering event. The differential cross-section for inelastic scattering can be expressed as:

  -------------------------------------------------------- [3b]

where,
           I₀​ is the incident electron intensity.
           Γ_k​ is the transition strength for the specific inelastic event (related to the oscillator strength of the transition).
           ω = E/ℏ is the energy loss divided by the reduced Planck constant.
           θ is the scattering angle.
           ΔE = E₀​ − E is the energy loss, with E₀​ being the initial energy (the energy of incident electron beam) and E the final energy (the energy of the electrons after penetrated through the TEM speciment).
          S(q,ω) is the dynamic structure factor, which depends on the momentum transfer q and energy loss ω.

In Equation 3b, the Γ_k/ω term represents the probability of the specific inelastic transition. Γ_k​ is related to the cross-section for the inelastic process and ω\omega ω is the energy loss (related to the frequency of the energy loss). The term cos²θ/(ΔE)² captures the angular dependence of the scattered intensity. cos²θ arises from the dipole approximation, which assumes that the inelastic scattering event is dipolar (as in many electron transitions). The (ΔE)² in the denominator emphasizes that lower energy losses are more probable, resulting in higher intensities at small ΔE (see Figure 3a). The S(q,ω) is a complex function that depends on the material's electronic structure and describes how the material responds to the inelastic scattering event. It is related to the density of states and the nature of the excitations (e.g., plasmons, interband transitions, etc.).

In solid angle, the differential cross-section for inelastic scattering of electrons in Formula 3a becomes:

  -------------------------------------------------------- [3ca]

where,
          dΩ is the differential solid angle.

The inelastic scattering process often involves transitions in the target material, such as electronic transitions, where an electron excites atoms within the sample. In the dipole approximation, the interaction between the incident electrons and the sample is assumed to be dominated by dipole transitions (electric dipoles), which is valid when the momentum transfer is small (i.e., small angles and energy losses).  Under this approximation, the double differential cross-section becomes:

  -------------------------------------------------------- [3cb]

Then, Equation 3b becomes:

  -------------------------------------------------------- [3cc]

In EELS, especially when using a Transmission Electron Microscope (TEM), the scattering angles are typically small (i.e., θ is small). Under this condition, the cosine term cos θ can be approximated as:

  -------------------------------------------------------- [3d]

Therefore, cos²θ ≈ 1 in TEM at small-angle scattering.

The dipole approximation assumes that the inelastic scattering process is dominated by dipole transitions, which are common in core-loss events. Under this approximation, the scattering cross-section is strongly peaked at small angles, which simplifies the angular dependence of the cross-section to a form that depends only on the angle θ and the characteristic angle θ_E​ .

The characteristic scattering angle θ_E is given by,

  -------------------------------------------------------- [3e]

The differential cross-section is then expressed as:

  -------------------------------------------------------- [3f]

where,
          The term E₀/E² reflects the dependence on the energy loss E = ΔE and the incident energy E₀.
          The factor describes how the scattering intensity depends on the scattering angle θ.

Combining the above elements, we obtain the Bethe Ridge formula for the differential cross-section: 

  -------------------------------------------------------- [3g]

The scattering intensity decreases with increasing energy loss E=ΔE, and the intensity is peaked at small scattering angles θ, with the characteristic angle θ_E determining the width of this peak. When θ is much smaller than θ_E, the term ​ approaches 1, meaning that the intensity remains high. As θ increases and becomes comparable to or larger than θ_E, the intensity falls off, reflecting the reduced probability of large-angle scattering.

Figure 3a shows the scattering intensity versus energy loss for different scattering angles based on Equation 3cc (Fundametal) and Equation 3f (Bethe Ridge).

Figure 3b shows the scattering intensity versus scattering angle for different energy losses based on Equation 3cc (Fundametal) and Equation 3f (Bethe Ridge).

The characteristic angle θ_E ​ in the Bethe Ridge formula is a key parameter that defines the angular distribution of the scattered electrons, which is given by: 

  -------------------------------------------------------- [3h]

Figure 3c shows the characteristic angles versus energy loss at different incident electron energies.