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.

Plural scattering

The inelastic mean free path \lambda 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.

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


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

  • \(I_{t}\) = total intensity

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

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


  • \(I_{o}\) = 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 \(I_{t}\).

Quantification graph

EELS processingYou can compute this easily from low-loss spectrum via the Thickness button located in the EELS processing palette in the Techniques panel. GMS 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. GMS 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 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.


Malis, T.; Cheng, S. C.; Egerton, R. F. EELS log ratio technique for specimen-thickness measurement in the TEM. J. Electron Microscope Technique. 8:193; 1988.

Iakoubovskii, K.; Mitsuishi, K.; Nakayama, Y.; Furuya, K. Thickness measurements with electron energy loss spectroscopy. Microscopy Research and Technique. 71(8):626 – 31; 2008.

Iakoubovskii, K.; Mitsuishi, K.; Nakayama, Y.; Furuya, K. Mean free path of inelastic electron scattering in elemental solids and oxides using transmission electron microscopy: Atomic number dependent oscillatory behavior. Physical Review B. 77(10):104102; 2008.

Egerton, R. F., Cheng, S. C. Measurement of local thickness by electron energy-loss spectroscopy. Ultramicroscopy. 21(3):231 – 244; 1987.