Plural Scattering and Sample Thickness

Plural scattering occurs when a significant fraction of incident electrons that pass through a sample are scattered inelastically more than once.

The inelastic mean free path represents the mean distance between inelastic scattering events for these electrons. When you regard inelastic scattering as a random event, the probability of n-fold inelastic scattering follows a Poisson distribution.

\(P_{n}=\frac{I_{n}}{I_{t}}=\frac{\left ( \frac{t}{\lambda } \right )^{n}}{n!}exp\left ( \frac{-t}{\lambda } \right )\)

where

• \(I_{n}\) = Intensity of n-fold scattering

• \(I_{t}\) = Total intensity

For \(n=0\), we get the simple relationship

\(t/\lambda = -ln( I_{o}/I_{t} )\)

where

• \(I_{o}\) = Intensity of 0-fold scattering (the elastic scattering signal)

This method is known as the log-ratio method. From a spectrum, the integral of the ZLP will give \(I_{o}\), while the integral of the entire spectrum will give \(I_{t}\).

You can compute this easily from the low-loss spectrum via the Thickness button located in the EELS processing palette in the Techniques panel. DigitalMicrograph will fit the ZLP to extract the \(I_{o}\) contribution. While the total spectrum is never truly measured, \(I_{t}\) can be approximated from a measured energy range since a majority of the signal is contained in the first 50 – 100 eV. DigitalMicrograph uses a power-law tail beyond the measured spectrum to estimate its contribution to \(I_{t}\).

When you perform this analysis, you can determine if the region is thin enough for EELS and if plural scattering has a significant impact.

Typically, just the ratio of \(t/\lambda\) is needed, but methods are available to estimate the value of the mfp for your material. Another method to determine the absolute thickness of the sample is the Kramers-Kronig Sum Rule.