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Constant-wavelength peak profiles

Use \(\Delta=2\theta-2\theta_h\) and FWHM \(H\) in degrees. Widths, mixing and the production asymmetry coefficients are evaluated at the reflection's Bragg angle, not independently at every sampled pattern point.

Symmetric primitives

\[G(\Delta;H)=\frac{2}{H}\sqrt{\frac{\ln2}{\pi}}\exp[-4\ln2(\Delta/H)^2],\qquad L(\Delta;H)=\frac{2}{\pi H}\frac{1}{1+4(\Delta/H)^2}.\]
\[pV(\Delta;H,\eta)=(1-\eta)G(\Delta;H)+\eta L(\Delta;H).\]

Both components use the same \(H\). cwl-gaussian uses \(\eta=0\) and cwl-lorentzian uses \(\eta=1\).

Direct width and mixing

For Gaussian, Lorentzian and direct pseudo-Voigt types,

\[H^2=U\tan^2\theta+V\tan\theta+W.\]

cwl-pseudo-voigt additionally uses \(\eta=\eta_0+\eta_1(2\theta)\) with \(2\theta\) in degrees. mixing_eta_1 is therefore per degree. An unphysical width or mixing choice is not repaired by interpreting the parameters as a different profile type.

Thompson–Cox–Hastings

cwl-tch-pseudo-voigt derives a combined width and mixing from

\[H_G^2=U\tan^2\theta+V\tan\theta+W,\qquad H_L=X\tan\theta+Y/\cos\theta.\]
\[\begin{aligned} H^5={}&H_G^5+2.69269H_G^4H_L+2.42843H_G^3H_L^2\\ &+4.47163H_G^2H_L^3+0.07842H_GH_L^4+H_L^5,\\ \eta={}&1.36603r-0.47719r^2+0.11116r^3,\qquad r=H_L/H. \end{aligned}\]

\(U,V,W\) have degree-squared units; \(X,Y\) have degree units. Here X multiplies \(\tan\theta\) and Y multiplies \(1/\cos\theta\). The implementation's nonnegative TCH mixing floor is zero, preserving the Gaussian endpoint. The combined \(H\) is used by both Gaussian and Lorentzian components.

Bérar–Baldinozzi asymmetry

cwl-pseudo-voigt-berar-baldinozzi applies the direct pseudo-Voigt width/mixing with an empirical multiplicative factor. Let \(z=\Delta/H\):

\[F_a(z)=2ze^{-z^2},\qquad F_b(z)=2(2z^2-3)F_a(z),\]
\[A(\theta)=A_0/\tan\theta+A_1/\tan2\theta,\qquad B(\theta)=B_0/\tan\theta+B_1/\tan2\theta,\]
\[\Omega(\Delta)=pV(\Delta)[1+A(\theta)F_a(z)+B(\theta)F_b(z)].\]

Parameters are asym_beba_a0, asym_beba_b0, asym_beba_a1, asym_beba_b1 and the limit angle asym_beba_limit. At or above that limit in degrees \(2\theta\), the asymmetry correction is omitted. It is a descriptor, not an independent free fit parameter. The production per-reflection evaluation keeps the odd factor centred on each reflection.

Finger–Cox–Jephcoat asymmetry

cwl-tch-pseudo-voigt-fcj convolves the TCH core with an axial-divergence weight. Put \(S_L=S/L\) and \(D_L=D/L\), half the sample and detector heights divided by the diffractometer radius:

\[\cos2\phi_{\min}=\cos2\theta_h\sqrt{1+(S_L+D_L)^2},\qquad \cos2\phi_{\rm infl}=\cos2\theta_h\sqrt{1+(S_L-D_L)^2}.\]

With \(q(\delta)=|\cos2\theta_h|/\sqrt{\cos^2\delta-\cos^22\theta_h}\), the two geometric regions use

\[G(\delta)=\begin{cases}\left|[(S_L+D_L)q(\delta)-1]/\cos\delta\right|,&\text{inner region},\\ 2\min(S_L,D_L)q(\delta)/\cos\delta,&\text{outer region}. \end{cases}\]

Integrate between \(2\phi_{\min}\) and \(2\theta_h\). The inner branch applies when \(|\delta-2\phi_{\min}|\le|2\phi_{\rm infl}-2\phi_{\min}|\); the outer branch applies beyond that boundary. The normalised quadrature is

\[\Omega(x)=\frac{\sum_k w_kG(\delta_k)\,pV(x-\delta_k+2\theta_h)}{\sum_k w_kG(\delta_k)}.\]

asym_fcj_1 is \(S_L\) and asym_fcj_2 is \(D_L\). The implementation handles zero/equal-height limits and uses the symmetric profile within 0.4° of \(2\theta_h=90°\); these evaluation conventions are part of the model, not new free parameters.

References

  1. Caglioti, G., Paoletti, A. & Ricci, F. P. (1958). Nuclear Instruments, 3, 223–228. Angular dependence of peak widths.
  2. Thompson, P., Cox, D. E. & Hastings, J. B. (1987). Journal of Applied Crystallography, 20, 79–83. DOI. TCH width and mixing.
  3. Bérar, J.-F. & Baldinozzi, G. (1993). Journal of Applied Crystallography, 26, 128–129. Empirical profile asymmetry.
  4. Finger, L. W., Cox, D. E. & Jephcoat, A. P. (1994). Journal of Applied Crystallography, 27, 892–900. Axial-divergence asymmetry.