3. Peak profiles

3.1. The instrument ⊕ sample width split

The Gaussian and Lorentzian component widths each carry an instrument part and a sample part, combined by the physics of convolution: Gaussian variances add, Lorentzian FWHMs add.

(3.1)\[\Gamma_G^2(\theta) \;=\; (U + U_s)\tan^2\theta + V\tan\theta + W + \frac{P}{\cos^2\theta} \qquad [\deg^2 2\theta]\]

Source: rietx.model.profiles.caglioti

\(U, V, W\) are the instrument resolution function [CPR58]; the sample adds a Gaussian microstrain term \(U_s\tan^2\theta\) and a Gaussian size term \(P/\cos^2\theta\) [LVD04, TCH87].

(3.2)\[\Gamma_L(\theta) \;=\; \frac{X + X_s}{\cos\theta} + (Y + Y_s)\tan\theta \qquad [\deg 2\theta]\]

Source: rietx.model.profiles.caglioti

Warning

Conventions here are documented by physics, not letters: the \(1/\cos\theta\) (and \(1/\cos^2\theta\) variance) terms carry Scherrer crystallite-size broadening, the \(\tan\theta\) (and \(\tan^2\theta\)) terms microstrain. The letter assignments differ between codes — GSAS uses X = size, Y = strain; FullProf swaps them. Transfer values by matching the \(\theta\)-law, never the letter.

Anisotropic (hkl-dependent) sample broadening is deferred to Microstructure.

3.2. Thompson-Cox-Hastings pseudo-Voigt

The default profile approximates the Voigt (Gaussian ⊗ Lorentzian) as a linear blend with a single FWHM \(\Gamma\) and mixing \(\eta\) [TCH87]:

(3.3)\[\mathrm{pV}(x) \;=\; \eta\, L(x; \Gamma) + (1 - \eta)\, G(x; \Gamma),\]

(3.4)\[\Gamma^5 = \Gamma_G^5 + 2.69269\, \Gamma_G^4 \Gamma_L + 2.42843\, \Gamma_G^3 \Gamma_L^2 + 4.47163\, \Gamma_G^2 \Gamma_L^3 + 0.07842\, \Gamma_G \Gamma_L^4 + \Gamma_L^5,\]

(3.5)\[\eta = 1.36603\, q - 0.47719\, q^2 + 0.11116\, q^3, \qquad q = \Gamma_L / \Gamma.\]

Source: rietx.model.profiles.pseudovoigt

Both component shapes are unit-area normalised, so \(\int \mathrm{pV}\,dx = 1\) and the reflection intensity of (1.2) enters purely through the prefactor:

(3.6)\[G(x) = \frac{2}{\Gamma}\sqrt{\frac{\ln 2}{\pi}} \exp\!\left(-\frac{4 \ln 2\, x^2}{\Gamma^2}\right), \qquad L(x) = \frac{2/(\pi\Gamma)}{1 + 4x^2/\Gamma^2}.\]

Source: rietx.model.profiles.pseudovoigt

3.3. The true Voigt, via the Faddeeva function

An opt-in shape (Instrument.profile.shape = "voigt"; TCHZ stays the default) evaluates the exact convolution [Arm67]:

(3.7)\[V(x; \sigma, \gamma) \;=\; \frac{\operatorname{Re}[w(z)]}{\sigma\sqrt{2\pi}}, \qquad z = \frac{x + i\gamma}{\sigma\sqrt{2}},\]

Source: rietx.model.profiles.voigt

where \(\sigma\) is the Gaussian standard deviation and \(\gamma\) the Lorentzian half-width at half maximum, recovered from the same component FWHMs of (3.1)-(3.2):

(3.8)\[\sigma = \frac{\Gamma_G}{2\sqrt{2\ln 2}}, \qquad \gamma = \frac{\Gamma_L}{2}.\]

Source: rietx.model.profiles.voigt

Both limits are exact and recovered branchlessly: \(\gamma \to 0\) makes \(z\) real and \(\operatorname{Re}[w] = e^{-z^2}\) (the unit Gaussian); \(\sigma \to 0\) sends \(|z| \to \infty\) where \(w(z) \to i/(\sqrt{\pi} z)\) (the unit Lorentzian).

The Faddeeva function \(w(z) = e^{-z^2}\operatorname{erfc}(-iz)\) (\(\operatorname{Im} z \ge 0\)) is computed by the Weideman rational approximation [Wei94]: the conformal map of the upper half-plane onto the unit disc,

(3.9)\[Z = \frac{L + iz}{L - iz}, \qquad L = \sqrt[4]{1/2}\cdot\sqrt{N}, \qquad w(z) = \frac{2\, p(Z)}{(L - iz)^2} + \frac{1/\sqrt{\pi}}{L - iz},\]

Source: rietx.model.profiles.faddeeva

with \(p\) an \(N\)-term polynomial whose real coefficients come from a single FFT at import time; \(N = 32\) reaches ≈1e-13. The algorithm was chosen for being branchless over the whole upper half-plane — no region partition (Humlíček’s w4 [Humlivcek82]) and no series switching [ZA11] — which keeps the residual smooth for finite-difference and autodiff Jacobians. The Voigt argument always has \(\operatorname{Im} z = \gamma_L/(\sigma\sqrt{2}) \ge 0\), so the reflection formula for the lower half-plane is never needed. Derivatives reuse the same \(w\) call through the identity \(w'(z) = -2z\,w(z) + 2i/\sqrt{\pi}\) [AS64].

3.4. Finger-Cox-Jephcoat axial divergence

With a sample of axial half-length \(S\) and a receiving slit of axial half-length \(H\) at goniometer radius \(L\), rays leaving the diffraction plane are detected at an apparent angle \(2\varphi\) related to the true Bragg angle by [FCJ94]

(3.10)\[\cos 2\varphi \;=\; \cos 2\theta \cdot \sqrt{1 + \xi^2}, \qquad \xi = u/L,\]

Source: rietx.model.profiles.fcj

where \(u\) is the signed axial offset of the ray. For \(2\theta < 90°\) intensity smears from \(2\theta\) down to \(2\varphi_{\min}\) — the classic low-angle tail of laboratory data; above 90° the smear is toward high angle. The weight of a given offset is the axial overlap of sample and slit, a trapezoid in \(\xi\) (with \(s = S/L\), \(h = H/L\)):

(3.11)\[W(\xi) \;=\; \operatorname{clip}\bigl(s + h - \xi,\; 0,\; 2\min(s, h)\bigr), \qquad \xi \ge 0.\]

Source: rietx.model.profiles.fcj

Expressed as a density in \(2\varphi\), the aberration diverges like \(1/\sqrt{2\theta - 2\varphi}\) at the Bragg position — the reason naive sampling fails. Substituting \(\xi\) as the integration variable removes the singularity exactly:

(3.12)\[y(2\theta_i) \;=\; \frac{\int_0^{\xi_{\max}} W(\xi)\, \Omega\bigl(2\theta_i - 2\varphi(\xi)\bigr)\, d\xi} {\int_0^{\xi_{\max}} W(\xi)\, d\xi}, \qquad \xi_{\max} = \min\bigl(s + h,\; |\tan 2\theta|\bigr),\]

Source: rietx.model.profiles.fcj

with a smooth integrand, evaluated by fixed-node Gauss-Legendre quadrature in \(\tau = \xi/\xi_{\max}\). The \(|\tan 2\theta|\) cap removes the unphysical \(\cos 2\varphi > 1\) branch at very low angle, and the weights are renormalised to \(\sum\omega = 1\) so the composite peak keeps exactly unit area — reflection intensities remain areas. Node counts are frozen per stage (6.0 nodes per FWHM of smear where the map moves fastest, clamped to [8, 64]); node positions follow \(s\), \(h\) and \(2\theta\) smoothly. When the asymmetric extent is below 0.02 of the peak FWHM the aberration is invisible and the peak is treated as symmetric.

The quadrature is split at the kink of the trapezoid (3.11), which keeps the response \(C^1\) everywhere except the inherent FCJ corner at \(s = h\) — a genuine non-differentiability with measured consequences for refinement when both apertures are equal, worked through in Reading a paper against its own numbers.