# 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 $\lambda$ 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.

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

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 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.

## References

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.