Thompson-cox-hastings Pseudo-voigt Function Apr 2026

[ \Gamma_V = \left( \Gamma_G^5 + 2.69269 , \Gamma_G^4 \Gamma_L + 2.42843 , \Gamma_G^3 \Gamma_L^2 + 4.47163 , \Gamma_G^2 \Gamma_L^3 + 0.07842 , \Gamma_G \Gamma_L^4 + \Gamma_L^5 \right)^1/5 ]

[ \eta = 1.36603 \left( \frac\Gamma_L\Gamma_V \right) - 0.47719 \left( \frac\Gamma_L\Gamma_V \right)^2 + 0.11116 \left( \frac\Gamma_L\Gamma_V \right)^3 ]

The is:

# Normalized Gaussian & Lorentzian with FWHM = G_V sigma = G_V / (2*sqrt(2*log(2))) # Gaussian sigma gamma = G_V / 2 # Lorentzian half-width at half-max t = (x - x0)

G = (1/(sigma*sqrt(2*pi))) * exp(-0.5*(t/sigma)**2) L = (gamma/pi) / (t**2 + gamma**2)

[ V_\textTCH(x) = \eta , L(x) + (1-\eta) , G(x) ]

return (1-eta)*G + eta*L P. Thompson, D. E. Cox, and J. B. Hastings, “Rietveld refinement of Debye–Scherrer synchrotron X‑ray data from Al₂O₃” , J. Appl. Cryst. (1987), 20, 79–83. The empirical FWHM and mixing coefficient formulas are given there.