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veight
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| May 15 2011 L.A. Times Sunday |
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In spectroscopy, the Voigt profile (named after Woldemar Voigt) is a line profile resulting 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. Voigt profiles are common in many branches of spectroscopy and diffraction. Due to the computational expense of the convolution operation, the Voigt profile is often approximated using a pseudo-Voigt profile. * All normalized line profiles can be considered to be probability distributions. The Gaussian profile has a Gaussian, or normal, distribution and a Lorentzian profile has a Lorentz, or Cauchy, distribution. Without loss of generality, we can consider only centered profiles, which peak at zero. The Voigt profile is then a convolution of a Lorentz profile and a Gaussian profile:* * * * V * ( * x * ; * σ * , * γ * ) * = * * ∫ * * − * ∞ * * * ∞ * * * G * ( * * x * ′ * * ; * σ * ) * L * ( * x * − * * x * ′ * * ; * γ * ) * * d * * x * ′ * * , * * * {\displaystyle V(x;\sigma ,\gamma )=\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 * * * ) * * * * σ * * * 2 * π * * * * * * , * * * {\displaystyle G(x;\sigma )\equiv {\frac {e^{-x^{2}/(2\sigma ^{2})}}{\sigma {\sqrt {2\pi }}}},} * and * * * * L * ( * x * ; * γ * ) * * * {\displaystyle L(x;\gamma )} * is the centered Lorentzian profile: * * * * * L * ( * x * ; * γ * ) * ≡ * * * γ * * π * ( * * x * * 2 * * * ... |