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

Source: rietx.model.forward

Each emission line (Kα₁/Kα₂, …) diffracts at its own Bragg angle, so the doublet splitting grows with \(\tan\theta\) — it is never a fixed \(2\theta\) offset (see (2.3)). The line weight \(w_l\) is the intensity of line \(l\) relative to line 0, which is structurally locked at 1 because it is degenerate with the phase scales. \(\Omega_{lk}\) is a unit-area profile (chapter Peak profiles), so the reflection intensity \(I_{pk}\) enters purely as an area.

1.1. Three intensity models

Rietveld mode computes intensities from the structural model [Rie69]:

(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 \(|F|^2\) and the Lorentz-polarisation factor Lp (chapter Intensity corrections). \(|F|^2\) depends only on \(\sin\theta/\lambda = 1/2d\) and is therefore shared across emission lines; Lp is evaluated per line.

Le Bail mode [LBDF88] treats the \(I_{pk}\) as empirical per-\(hkl\) values, 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

which is a fixed point when \(y_{\mathrm{obs}} = y_{\mathrm{calc}}\). The extracted intensities live outside the parameter vector and are path-dependent, so history nodes serialize them per node rather than treating them as parameters.

Pawley mode [Paw81] instead places the per-\(hkl\) intensities inside the least-squares problem, as an off-table parameter block appended to \(\theta\). Equation (1.3) is then used exactly 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 reported at ≈100 % uncertainty rather than with the spuriously tight esd a bare pseudo-inverse of a singular \(J^\top J\) would give. Such groups come back flagged PAWLEY_OVERLAP_UNRESOLVED.

1.2. The residual row layout

The least-squares residual is not just the data block. Its row layout,

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

Source: rietx.model.rows

is defined once, in rietx.model.rows, and every builder — the numpy residual, the numpy Jacobian’s row offsets, and the traced jax/torch residuals — consumes it. 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 restraints (Parameterisation and constraints).

1.3. Discreteness is frozen per stage

Everything discrete about the model — the reflection list, per-atom symmetry-operator subsets, the per-(line, reflection) evaluation windows (which extend ±(30.0 · estimated FWHM + a floor + the FCJ smear extent)), and the FCJ quadrature node counts — is computed when a stage is compiled and never changes during a least-squares run. Only node positions and weights follow the parameters, smoothly. This is what keeps the residual smooth for finite-difference and autodiff Jacobians; regeneration happens between stages only.