Practical Electron Microscopy and Database

For the single scattering spectrum, the overall intensity of the spectrum can be given by,
          I(E) = AE^-r + p₁σ₁(E) + p₂σ₂(E) + ... + p_Nσ_N(E) -------- [956a]
where,
         p₁~ p_N -- The probability of scatterings,
         σ₁~ σ_N -- The cross sections from different elements.
         AE^-r -- Power-law background fit.

Equation 956a can be re-written to represent the spectrum as a power-law background and core-loss profile for each edge:
          I(E) = AE^-r + C₁I_C1(E) + C₂I_C2(E) + ... + C_NI_CN(E) -------- [956b]
where,
         C₁ - C_N -- Weightings of the contribution of each element.

In EELS measurements, the single scattering spectrum can be approximately given by, [1]
           ------------------------ [956c]
where,
          , for θ ≤ θc ------------------------ [956d]
          , for θ > θc ------------------------- [956e]
          , which is the critical angle for inner-shell excitation, [2] and is in practice an angle in the range of 5 to 6 mrad (this is normally greater than the angular acceptance of the objective aperture).
           θ_E = E/(2E₀)

The step-function in Equations 956d and 956e is introduced to simplify the (weak) angular dependence of the oscillator strength on scattering angle.

Then, the single scattering into an aperture of semi-angle β can be given by,
           ------------------------ [956f]

The EELS signal from single scattering can be modeled using Poisson Distribution.

[1] C. Colliex, V. E. Cosslett, R.D.Leapman, P.Trebbia, Contribution of electron energy loss spectroscopy to the development of analytical electron microscopy, 1 (3–4), (1976), 301-315.
[2] N.F. Mott and H.S.W. Massey, Theory of Atomic Collisions (Clarendon Press, 1968).