The Voigt profile (named after Woldemar Voigt) is a probability distribution given by a convolution of a Cauchy-Lorentz distribution and a Gaussian distribution. It is often used in analyzing data from spectroscopy or diffraction.
Definition Without loss of generality, we can consider only centered profiles, which peak at zero. The Voigt profile is then
V ( x ; σ , γ ) ≡ ∫ − ∞ ∞ G ( x ′ ; σ ) L ( x − x ′ ; γ ) d x ′ , {\displaystyle V(x;\sigma ,\gamma )\equiv \int _{-\infty }^{\infty }G(x';\sigma )L(x-x';\gamma )\,dx',}
where x is the shift from the line center, G ( x ; σ ) {\displaystyle G(x;\sigma )} is the centered Gaussian profile:
G ( x ; σ ) ≡ e − x 2 2 σ 2 2 π σ , {\displaystyle G(x;\sigma )\equiv {\frac {e^{-{\frac {x^{2}}{2\sigma ^{2}}}}}{{\sqrt {2\pi }}\,\sigma }},}
and L ( x ; γ ) {\displaystyle L(x;\gamma )} is the centered Lorentzian profile:
L ( x ; γ ) ≡ γ π ( γ 2 + x 2 ) . {\displaystyle L(x;\gamma )\equiv {\frac {\gamma }{\pi (\gamma ^{2}+x^{2})}}.}
The defining integral can be evaluated as:
V ( x ; σ , γ ) = Re [ w ( z ) ] 2 π σ , {\displaystyle V(x;\sigma ,\gamma )={\frac {\operatorname {Re} [w(z)]}{{\sqrt {2\pi }}\,\sigma }},}
where Re[w(z)] is the real part of the Faddeeva function evaluated for
z = x + i γ 2 σ . {\displaystyle z={\frac {x+i\gamma }{{\sqrt {2}}\,\sigma }}.}
In the limiting cases of σ = 0 {\displaystyle \sigma =0} and γ = 0 {\displaystyle \gamma =0} then V ( x ; σ , γ ) {\displaystyle V(x;\sigma ,\gamma )} simplifies to L ( x ; γ ) {\displaystyle L(x;\gamma )} and G ( x ; σ ) {\displaystyle G(x;\sigma )} , respectively.
History and applications In spectroscopy, a Voigt profile results from the convolution of two broadening mechanisms, one of which alone would produce a Gaussian profile (usually, as a result of the Doppler broadening), and the other would produce a Lorentzian profile (damping of the emission or absorption, as considered in the original article by Waldemar Voigt 1912). Voigt profiles are common in many branches of spectroscopy and diffraction. Due to the expense of computing the Faddeeva function, the Voigt profile is sometimes approximated using a pseudo-Voigt profile.
Properties The Voigt profile is normalized:
∫ − ∞ ∞ V ( x ; σ , γ ) d x = 1 , {\displaystyle \int _{-\infty }^{\infty }V(x;\sigma ,\gamma )\,dx=1,}
since it is a convolution of normalized profiles. The Lorentzian profile has no moments (other than the zeroth), and so the moment-generating function for the Cauchy distribution is not defined. It follows that the Voigt profile will not have a moment-generating function either, but the characteristic function for the Cauchy distribution is well defined, as is the characteristic function for the normal distribution. The characteristic function for the (centered) Voigt profile will then be the product of the two:
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