1. The forward model

A constant-wavelength powder pattern is modelled as a background plus a triple sum over phases \(p\), source emission lines \(l\) and reflections \(k\):

(1.1)\[y_{\mathrm{calc}}(2\theta_i) \;=\; y_{\mathrm{bkg}}(2\theta_i) \;+\; \sum_p \sum_l \sum_k I_{pk}\, w_l\, \Omega_{lk}(2\theta_i) \qquad [\text{counts}]\]

Source: rietx.model.forward

Each emission line (Kα₁/Kα₂, …) diffracts at its own Bragg angle, so the doublet splitting grows with \(\tan\theta\), as in (2.5). It is not a fixed \(2\theta\) offset. The line weight \(w_l\) is the intensity of line \(l\) relative to line 0. Line 0 itself is locked at 1, because its weight is degenerate with the phase scales.

The three factors carry the units of Symbols and units. \(\Omega_{lk}\) is a unit-area profile (chapter Peak profiles), so it is a density on the angle axis, in deg⁻¹. \(w_l\) is a ratio. The reflection intensity \(I_{pk}\) is therefore an area, in counts·deg 2θ: what the line integrates to, and not the height it reaches. That is the conversion a reader coming from another code needs most often.

1.1. Three intensity models

1.1.1. Rietveld mode

Intensities come from the structural model [Rietveld, 1969]:

(1.2)\[I_{pk} \;=\; S_p \cdot m_{pk} \cdot |F_{pk}|^2 \cdot \mathrm{Lp}(2\theta_{lk}),\]

Source: rietx.model.forward

with phase scale \(S_p\), multiplicity \(m_{pk}\) (chapter Intensities), structure factor \(|F|^2\) in e² and the Lorentz-polarisation factor Lp (chapter Intensity corrections). Everything but \(S_p\) is fixed by the model, so the scale is what turns e² into the counts·deg 2θ of \(I_{pk}\). Its value carries no meaning on its own. It is meaningful against the other phases’ scales, and the quantitative fractions of (5.16) are therefore ratios. \(|F|^2\) depends only on \(\sin\theta/\lambda = 1/2d\) and is shared across emission lines. Lp is evaluated per line.

1.1.2. Le Bail mode

Intensities are empirical per-\(hkl\) values [Le Bail et al., 1988], updated between least-squares cycles by observed-intensity partitioning summed over lines:

(1.3)\[I_k \;\leftarrow\; \frac{\sum_l \sum_i \bigl[I_k\, w_l\, \Omega_{lk,i} / y_{\mathrm{bragg},i}\bigr] \cdot \max(y_{\mathrm{obs},i} - y_{\mathrm{bkg},i},\, 0)} {\sum_l w_l \sum_i \Omega_{lk,i}},\]

Source: rietx.model.forward.CompiledModel.lebail_update

The update is a fixed point when \(y_{\mathrm{obs}} = y_{\mathrm{calc}}\). The extracted intensities live outside the parameter vector and are path-dependent, so a history node serializes them beside the parameters it stores.

1.1.3. Pawley mode

The per-\(hkl\) intensities sit inside the least-squares problem [Pawley, 1981], as an off-table parameter block appended to \(\theta\). Equation (1.3) is then used once, to seed the block before the first solve. Reflections whose primary-line centres sit within 0.5 × their mean FWHM form an overlapped group and receive a soft equal-split restraint, scaled so that the split-direction esd is of order the group intensity itself. An unresolved split is then reported at ≈100 % uncertainty, where a bare pseudo-inverse of a singular \(J^\top J\) would report a spuriously tight one. Such groups come back flagged PAWLEY_OVERLAP_UNRESOLVED.

The reflection list runs 0.5° past each end of the fitted window, so that a reflection a stage moves onto the data is modelled. A reflection with no emission line centred on the data reaches it only through a tail, so its column is nearly zero and its intensity is bounded below only. Each such reflection gets one more row in the same block, a ridge toward zero,

(1.4)\[r_k \;=\; \frac{\sqrt{\lambda}}{s_p}\, I_k, \qquad s_p = \max_{j\,\text{on the data}} I_j ,\]

Source: rietx.model.forward.CompiledModel.build_pawley_restraint

where \(s_p\) is the phase’s largest on-data intensity when the stage starts. An intensity the data do not determine then stays within \(s_p\), with an esd of that order. One the data do reach has a column far more precise than a prior that wide. Without the ridge, a synthetic series refined one such intensity to \(10^{12}\), and TRF’s step test, which is relative to \(\lVert x\rVert\), ended warm refits early (WP-1459). The result names the ridged reflections, one PAWLEY_OFF_DATA_RIDGED diagnostic per phase, because their intensities are the ridge’s and not the data’s.

1.2. The residual row layout

The residual carries four blocks of rows, and the data is the first. The layout,

(1.5)\[r \;=\; \bigl[\; \text{data} \;\big|\; \text{background penalty} \;\big|\; \text{Pawley restraint} \;\big|\; \text{soft restraint} \;\bigr],\]

Source: rietx.model.rows

is defined once, in rietx.model.rows. Every builder consumes it: the numpy residual, the numpy Jacobian’s row offsets, and the traced jax/torch residuals. The data rows are \(\sqrt{w_i}\,(y_{\mathrm{obs},i} - y_{\mathrm{calc},i})\). The remaining blocks are described with the background models (Background) and the restraints (Parameterisation and constraints).

1.3. Discreteness frozen per stage

Four things are computed when a stage is compiled and never change during a least-squares run: the reflection list, the per-atom symmetry-operator subsets, the per-(line, reflection) evaluation windows, and the FCJ quadrature node counts. A window extends ±(k(η) · estimated FWHM + a floor + the FCJ smear extent), with k(η) sized so that the discarded pseudo-Voigt area stays at or below 0.02 (rietx.model.forward.window_fwhm_mult).

Node positions and weights follow the parameters, smoothly. Freezing everything else keeps the residual smooth for finite-difference and autodiff Jacobians. Regeneration happens between stages.