# Backscattering and absorption coefficients for electrons: Solutions of invariant embedding transport equations using a method of convergence

## Abstract

The backscattering coefficient is a magnitude whose measurement is fundamental for the characterization of materials with techniques that make use of particle beams and particularly when performing microanalysis. In this work, we report the results of an analytic method to calculate the backscattering and absorption coefficients of electrons in similar conditions to those of electron probe microanalysis. Starting on a five level states ladder model in 3D, we deduced a set of integro-differential coupled equations of the coefficients with a method know as invariant embedding. By means of a procedure proposed by authors, called method of convergence, two types of approximate solutions for the set of equations, namely complete and simple solutions, can be obtained. Although the simple solutions were initially proposed as auxiliary forms to solve higher rank equations, they turned out to be also useful for the estimation of the aforementioned coefficients. In previous reports, we have presented results obtained with the complete solutions. In this paper, we present results obtained with the simple solutions of the coefficients, which exhibit a good degree of fit with the experimental data. Both the model and the calculation method presented here can be generalized to other techniques that make use ofmore »

- Authors:

- Laboratorio de Física del Sólido, Dpto. de Física, Facultad de Ciencias Exactas y Tecnología, Universidad Nacional de Tucumán (Argentina)

- Publication Date:

- OSTI Identifier:
- 22275520

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Journal of Applied Physics; Journal Volume: 115; Journal Issue: 19; Other Information: (c) 2014 AIP Publishing LLC; Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND SUPERFLUIDITY; APPROXIMATIONS; BACKSCATTERING; CONVERGENCE; ELECTRON MICROPROBE ANALYSIS; ELECTRONS; INTEGRO-DIFFERENTIAL EQUATIONS; MATERIALS TESTING; MATHEMATICAL SOLUTIONS; PARTICLE BEAMS; TRANSPORT THEORY

### Citation Formats

```
Figueroa, C., Brizuela, H., and Heluani, S. P.
```*Backscattering and absorption coefficients for electrons: Solutions of invariant embedding transport equations using a method of convergence*. United States: N. p., 2014.
Web. doi:10.1063/1.4876124.

```
Figueroa, C., Brizuela, H., & Heluani, S. P.
```*Backscattering and absorption coefficients for electrons: Solutions of invariant embedding transport equations using a method of convergence*. United States. doi:10.1063/1.4876124.

```
Figueroa, C., Brizuela, H., and Heluani, S. P. Wed .
"Backscattering and absorption coefficients for electrons: Solutions of invariant embedding transport equations using a method of convergence". United States. doi:10.1063/1.4876124.
```

```
@article{osti_22275520,
```

title = {Backscattering and absorption coefficients for electrons: Solutions of invariant embedding transport equations using a method of convergence},

author = {Figueroa, C. and Brizuela, H. and Heluani, S. P.},

abstractNote = {The backscattering coefficient is a magnitude whose measurement is fundamental for the characterization of materials with techniques that make use of particle beams and particularly when performing microanalysis. In this work, we report the results of an analytic method to calculate the backscattering and absorption coefficients of electrons in similar conditions to those of electron probe microanalysis. Starting on a five level states ladder model in 3D, we deduced a set of integro-differential coupled equations of the coefficients with a method know as invariant embedding. By means of a procedure proposed by authors, called method of convergence, two types of approximate solutions for the set of equations, namely complete and simple solutions, can be obtained. Although the simple solutions were initially proposed as auxiliary forms to solve higher rank equations, they turned out to be also useful for the estimation of the aforementioned coefficients. In previous reports, we have presented results obtained with the complete solutions. In this paper, we present results obtained with the simple solutions of the coefficients, which exhibit a good degree of fit with the experimental data. Both the model and the calculation method presented here can be generalized to other techniques that make use of different sorts of particle beams.},

doi = {10.1063/1.4876124},

journal = {Journal of Applied Physics},

number = 19,

volume = 115,

place = {United States},

year = {Wed May 21 00:00:00 EDT 2014},

month = {Wed May 21 00:00:00 EDT 2014}

}