## Spectra of Electrons Emerging from PMMA

### Monte Carlo Simulation of Electron Energy Distributions

- Fig. 1: Energy distribution of the electrons emerging from PMMA with energies between 0 and 20eV. Monte Carlo simulated spectrum (red solid line) is compared to the Joy et al. experimental spectrum [8] (black line). Data are normalized to a common maximum. The primary energy is 1000eV. The primary electron beam is normal to the surface. Electrons are accepted over an angular range from 36° to 48° integrated around the full 360° azimuth. The zero of the energy scale is located at the vacuum level.
- Fig. 2: Monte Carlo simulated spectrum of electrons emerging from PMMA with energies between 750eV and E0=800eV (E0 = primary electron energy). The plasmon-loss peak is located at about 22eV from the elastic peaks. The primary electron beam is normal to the surface. Electrons are accepted over an angular range from 0° to 90° integrated around the full 360° azimuth. The zero of the energy scale is located at the vacuum level.
- Fig 3: Monte Carlo simulated total electron yield σ = δ + η of PMMA as a function of the primary electron kinetic energy E0 (solid line). The Monte Carlo data were obtained integrating the curves of energy distribution including all the electrons emerging with energy from 0 to E0. Experimental data: taken from reference [9] (red circles) and from reference [10] (green triangles).

**This work describes a Monte Carlo algorithm which appropriately takes into account the stochastic behavior of electron transport in solids and treats event-by-event all the elastic and inelastic interactions between the incident electrons and the particles of the solid target. The energy distributions of secondary and backscattered electrons emerging from polymethylmethacrylate (PMMA) irradiated by an electron beam are simulated and compared to the available experimental data.**

**The Spectrum**

When an electron beam impinges on a solid target, many electrons can be backscattered, after they interacted with the atoms and electrons of the target. A fraction of them conserves their original kinetic energy, having suffered only elastic scattering collisions with the atoms of the target. These electrons constitute the so-called elastic peak, or zero-loss peak, whose maximum is located at the energy of the primary beam. Close to the elastic peak, another feature can be observed: it is a broad peak collecting all the electrons of the primary beam which suffered inelastic interactions with the outer-shell atomic electrons (plasmons losses, and inter-band and intra-band transitions). Another important feature of the electron energy spectrum is represented by the secondary-electron emission distribution, i.e., the energy distribution of those electrons that, once extracted from the atoms by inelastic collisions and having travelled in the solid, reach the surface with the energy sufficient to emerge. The energy distribution of the secondary electrons is mainly confined in the low energy region of the spectrum, typically well below 50eV [1].

**The Monte Carlo Algorithm**

The results presented in this paper were obtained using differential and total elastic scattering cross sections calculated utilizing Mott theory [2], i.e. numerically solving the Dirac equation in a central field; this procedure is known as the “relativistic partial wave expansion method” and it has been demonstrated to provide excellent results when compared to experimental data. On the side of the energy losses, the inelastic mean free paths are calculated by taking into account the inelastic interactions of the incident electrons with atomic electrons, phonons, and polarons.

The calculation of the electron-electron inelastic scattering processes was performed within the Mermin theory [3]. Electron–phonon interactions were described using the Fröhlich theory [4]. Polaronic effect was modeled according to the law proposed by Ganachaud and Mokrani [5]. Electron trajectories follow a stochastic process, with scattering events separated by straight paths having a distribution of lengths that follows a Poisson-type law. Once the step length is generated, the elastic or inelastic nature of the next scattering event, the polar and azimuthal angles, and the energy losses, are all sampled using the relevant cumulative probabilities according to the usual Monte Carlo recipes [6]. Details of the Monte Carlo calculations can be found in reference [7].

**Results**

The Monte Carlo energy distribution of the electrons emerging from PMMA irradiated by an electron beam with primary energy E0=1000eV is presented in figure 1. The spectrum is simulated assuming that the primary electron beam is normal to the surface. The zero of the energy is located at the vacuum level. In the same figure, a comparison of the Monte Carlo simulated spectrum to the Joy et al. experimental electron energy distribution [8] is shown. The same conditions of the experiment are used for the simulation, i.e. acceptance angles in the range from 36° to 48° integrated around the full 360° azimuth (Cylindrical Mirror Analyzer geometry). The Monte Carlo calculation describes very well the initial increase of the spectrum, the energy position of the maximum, and the full width at half maximum. Monte Carlo simulation is not able, on the other hand, to describe the fine structure of the peak, in particular the observed shoulder on the left of the maximum.

In figure 2 the spectrum of the electrons emerging close to the elastic peak (backscattered electrons) is presented. The plasmon loss peak is located at about 22eV from the zero loss peak.

In the experiments, on the one hand, the secondary electron emission yield δ is measured as the integral of the energy distribution of all the emitted electrons over the energy range from 0 to 50eV. The backscattering coefficient η, on the other hand, is measured integrating the distribution of all the emitted electrons over the energy range from 50eV to the energy of the elastic peak.

In figure 3 the Monte Carlo simulated total electron emission yield σ = δ + η is compared, in the primary electron energy from 0 to 1500eV, to the available experimental data [9,10].

**Conclusion**

We have described and used a Monte Carlo algorithm for the evaluation of the energy distributions of secondary and backscattered electrons from polymethylmethacrylate irradiated by an electron beam. The integration of the distributions over the appropriate energy ranges allows calculating the secondary electron yield, δ, the backscattering coefficient, η, and the total electron yield, σ, as a function of the primary electron energy. The Monte Carlo simulated data are in agreement to the available experimental data.

**Acknowledgments**

Warm thanks are due to Diego Bisero (University of Ferrara), Giovanni Garberoglio (ECT*-FBK, Trento) and Cornelia Rodenburg (University of Sheffield) for fruitful discussions and stimulating suggestions. This work was supported by Istituto Nazionale di Fisica Nucleare (INFN) through the Supercalcolo agreement with FBK.

**References**

[1] R. Cimino, I. R. Collins, M. A. Furman, M. Pivi, F. Ruggiero, G. Rumolo, and F. Zimmermann: Can Low-Energy Electrons Affect High-Energy Physics Accelerators?

Phys. Rev. Lett. 93, 014801 (2004) http://dx.doi.org/10.1103/PhysRevLett.93.014801

[2] N.F. Mott: The Scattering of Fast Electrons by Atomic Nuclei, *Proc. R. Soc. London Ser.* 124, 425 (1929)

[3] N.D. Mermin: Lindhard Dielectric Function in the Relaxation-Time Approximation, *Phys. Rev.* B 1, 2362 (1970)

[4] H. Frӧhlich, *Adv. Phys.* 3, 325 (1954)

[5] J.P. Ganachaud and A. Mokrani: Theoretical study of the secondary electron emission of insulating targets, *Surf. Sci*. 334, 329 (1995), doi: 10.1016/0039-6028(95)00474-2

[6] M. Dapor: Transport of Energetic Electrons in Solids: Computer Simulation with Applications to Materials Analysis and Characterization, Vol. 257 of Springer Tracts in Modern Physics, Springer, Berlin (2014), ISBN 978-3-319-03883-4

[7] M. Dapor: *Appl. Surf. Sci.* (2016) (in press)

[8] D.C. Joy, M.S. Prasadh, M. Meyer III: Experimental secondary electron spectra under SEM conditions, *Journal of Microscopy*, 215, 77, (2004) http://dx.doi.org/10.1111/j.0022-2720.2004.01345.x

[9] M. Boubaya and G. Blaise: Charging regime of PMMA studied by secondary electron emission, *Eur. Phys. J.: Appl. Phys*. 37, 79 (2007) http://dx.doi.org/10.1051/epjap:2006128

[10] E.N. Evstaf’eva, E.I. Rau and R.A. Sennov: Some Kinetic Aspects of the Charging of Dielectric Targets by a (1–50)-keV Electron Beam, *Phys. Solid State* 50, 599 (2008)

**Author
Dr. Maurizio Dapor**

European Centre for Theoretical Studies in Nuclear Physics and Related Areas (ECT*-FBK)

and Trento Institute for Fundamental Physics and Applications (TIFPA-INFN)

Povo, Trento, Italy