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Time-of-flight peak profiles

Let \(\delta=t-t_h\) in microseconds. The bank geometry supplies the wavelength of each reflection; calibration maps spacing to the peak centre.

Gaussian and Lorentzian width terms

\[\sigma_G^2(d)=v_0+(v_1+\mathrm{strain}_G)d^2+(v_2+\mathrm{size}_G)d^4,\]
\[H_G=\sqrt{8\ln2}\,\sigma_G,\qquad H_L(d)=\gamma_0+(\gamma_1+\mathrm{strain}_L)d+(\gamma_2+\mathrm{size}_L)d^2.\]

Here \(v_0,v_1,v_2\) denote the stored variance coefficients labelled σ₀², σ₁² and σ₂²; the superscript is part of the variance notation, not an instruction to square their stored values again. The broad_gauss_sigma_* coefficients are variance coefficients, not standard deviations or FWHMs. broad_lorentz_gamma_* coefficients define Lorentzian FWHM. Keep their respective time/spacing units; see the parameter index.

Symmetric pseudo-Voigt

tof-pseudo-voigt uses the TCH width/mixing, with \(H_G,H_L\) in microseconds, and the symmetric unit-area pseudo-Voigt. It does not add the back-to-back exponential pulse.

Jorgensen pulse and profile

tof-jorgensen convolves a two-sided exponential pulse with a Gaussian. The rates are

\[\alpha(d)=\alpha_0+\alpha_1/d,\qquad \beta(d)=\beta_0+\beta_1/d^4,\]

with positive evaluated rates in inverse microseconds. Let \(N=\alpha\beta/[2(\alpha+\beta)]\) and \(\sigma\) be the Gaussian standard deviation:

\[\Omega_G(\delta)=N[e^u\operatorname{erfc}(y)+e^v\operatorname{erfc}(z)],\]
\[\begin{aligned} u&=\tfrac12\alpha(\alpha\sigma^2+2\delta),& y&=(\alpha\sigma^2+\delta)/(\sqrt2\sigma),\\ v&=\tfrac12\beta(\beta\sigma^2-2\delta),& z&=(\beta\sigma^2-\delta)/(\sqrt2\sigma). \end{aligned}\]

The normalisation already includes the Gaussian-convolution factors; do not reuse \(N\) as if it alone normalised a bare piecewise exponential. The numerical evaluation uses stable exponential/error-function combinations to avoid overflow without changing this equation.

Jorgensen–Von Dreele

tof-jorgensen-von-dreele adds the Lorentzian contribution through a common TCH width \(H\) and mixing \(\eta\):

\[\Omega=(1-\eta)\Omega_G+\eta\Omega_L.\]

Both terms use that combined width: \(\sigma=H/\sqrt{8\ln2}\) in \(\Omega_G\) and FWHM \(H\) in \(\Omega_L\).

\[\Omega_L=-\frac{2N}{\pi}\left(\operatorname{Im}[e^pE_1(p)]+\operatorname{Im}[e^qE_1(q)]\right),\]
\[p=\alpha\delta+i\alpha H/2,\qquad q=-\beta\delta+i\beta H/2.\]

\(E_1\) is the exponential integral. At zero Lorentzian width the profile reduces to its Gaussian Jorgensen limit; assigning a different width to each mixed kernel would define a different model.

Ikeda–Carpenter

Planned. Ikeda–Carpenter is planned for TOF pulse modelling. It is not a selectable profile in this reference snapshot. See Time-of-flight peak profiles for the availability of this and the implemented profiles.

References

  1. Von Dreele, R. B., Jorgensen, J. D. & Windsor, C. G. (1982). Journal of Applied Crystallography, 15, 581–589. Time-of-flight profile modelling.
  2. Thompson, P., Cox, D. E. & Hastings, J. B. (1987). Journal of Applied Crystallography, 20, 79–83. DOI. Width and mixing convention.