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Analytical expression for the convolution of a Fano line profile with a Gaussian

S. Schippers

TL;DR

This work derives a closed-form analytic expression for the convolution of a Fano line profile with a Gaussian window using the Faddeeva function $w(z)$, enabling fast and accurate spectral peak fitting. By connecting the Fano-Gaussian convolution to the Voigt framework through $\Re[w(z)]$ and introducing an asymmetric term via $\Im[w(z)]$, the authors provide a practical formula: $C(E)=\frac{a}{q^2}\frac{2\sqrt{\ln 2}}{\Delta_G\sqrt{\pi}}\bigl\{(q^2-1)\Re[w(z)]-2q\Im[w(z)]\bigr\}$, with $z=x+iy$ and $x,y$ defined from the widths $\Delta_L$ and $\Delta_G$. This yields a direct route to model fitting in spectroscopy, preserves total line strength, and clarifies how asymmetry arises from the imaginary part of $w(z)$. The paper also discusses numerical implementations of the Faddeeva function and demonstrates broader applicability to related line shapes beyond the Fano-Gaussian case.

Abstract

Asymmetric Fano line profiles are frequently encountered, e.g., in the photoionization spectra of atoms and ions. For the fitting of spectral line profiles to experimental spectra the line profiles have to be convolved with the experimental window function. The latter is often taken to be a Gaussian. It is shown that the convolution can be represented by a rather simple analytic expression involving the Faddeeva function for the evaluation of which efficient and accurate numerical algorithms are available.

Analytical expression for the convolution of a Fano line profile with a Gaussian

TL;DR

This work derives a closed-form analytic expression for the convolution of a Fano line profile with a Gaussian window using the Faddeeva function , enabling fast and accurate spectral peak fitting. By connecting the Fano-Gaussian convolution to the Voigt framework through and introducing an asymmetric term via , the authors provide a practical formula: , with and defined from the widths and . This yields a direct route to model fitting in spectroscopy, preserves total line strength, and clarifies how asymmetry arises from the imaginary part of . The paper also discusses numerical implementations of the Faddeeva function and demonstrates broader applicability to related line shapes beyond the Fano-Gaussian case.

Abstract

Asymmetric Fano line profiles are frequently encountered, e.g., in the photoionization spectra of atoms and ions. For the fitting of spectral line profiles to experimental spectra the line profiles have to be convolved with the experimental window function. The latter is often taken to be a Gaussian. It is shown that the convolution can be represented by a rather simple analytic expression involving the Faddeeva function for the evaluation of which efficient and accurate numerical algorithms are available.

Paper Structure

This paper contains 5 sections, 23 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The Voigt profile
  3. Some properties of the Faddeeva function
  4. Convolution of a Fano profile with a gaussian
  5. Summary

Figures (1)

  • Figure 1: Real part (panel a) and imaginary part (panel b) of the Faddeeva function $w(x+iy)$ as function of the scaled energy $x = 2\sqrt{\ln 2}(E_{\rm res}-E)/\Delta_G$ for different ratios $y = \Delta_L\sqrt{\ln 2}/\Delta_G$ of Lorentzian and Gaussian widths. The real part is essentially the Voigt profile (Eq. \ref{['eq:voigt']}). The imaginary part occurs in addition in the convolution of a Fano profile with a Gaussian (Eq. \ref{['eq:c3']}).