4. Intensities¶

4.1. The species scattering factor¶

(4.1)¶\[f(k, \lambda) \;=\; f_0(k) + f'(\lambda) + i\, f''(\lambda), \qquad k = \frac{\sin\theta}{\lambda}\ [\text{Å}^{-1}].\]

Source: rietx.crystallography.dispersion

\(f_0\) is the angle-dependent, wavelength-independent elastic form factor; \(f'\) and \(f''\) are the angle-independent, wavelength-dependent dispersion corrections. The split is physical: \(f_0\) probes the whole electron density and is therefore keyed by ion (La3+), while \(f'/f''\) are core-level resonance effects, essentially independent of valence, keyed by element (La). \(f'' \ge 0\) in this convention.

\(f_0\) uses the five-Gaussian parameterisation of Waasmaier & Kirfel [WK95]:

(4.2)¶\[f_0(k) \;=\; \sum_{i=1}^{5} a_i\, e^{-b_i k^2} + c, \qquad \text{valid for } k \le 6\ \text{Å}^{-1}.\]

Source: rietx.crystallography.scattering

\(f'/f''\) come from the Cromer-Liberman tabulation [CL70, CL81] — the crystallographic reference calculation (what [Pri04] §4.2.6 tabulates), chosen so that a disagreement with another Rietveld code is attributable rather than mysterious — with the Kissel & Pratt high-energy-limit correction [KP90], which reaches −1.3 e at uranium. Absorption edges are never interpolated across (\(f''\) jumps by nearly an order of magnitude across one grid interval at the Fe K edge), and within 50.0 eV of an edge the request is refused outright: there the true \(f''\) is the XANES of the compound, which no atomic table knows. Measured overrides can be supplied instead. Dispersion is applied by default (it was opt-in through v0.6, so every number recorded in docs/milestones/ up to and including that milestone was measured without it). Setting source.dispersion = None declines it and reproduces those numbers bit-identically; DISPERSION_NEGLECTED then says so, because having declined a correction that needs nothing but the species and the wavelength is a modelling statement rather than a silence.

4.2. The structure factor¶

(4.3)¶\[F(hkl) \;=\; \sum_j \mathrm{occ}_j\, f_j(k) \sum_m T_{jm}(\mathbf{h})\, e^{2\pi i\, \mathbf{h}\cdot(R_m \mathbf{x}_j + \mathbf{t}_m)},\]

Source: rietx.crystallography.structure_factor

where the inner sum runs over a per-atom subset of symmetry operations, chosen once per stage so special-position images are not double counted (the subset is frozen — discrete — while the positions it produces remain smooth functions of the refined coordinates). Intensities use \(|F|^2\) with the reflection multiplicity applied separately [Rie69]; multiplicities are computed by explicit orbit counting under the Laue group, so \(\pm\mathbf{h}\) are always merged into one orbit.

4.3. Debye-Waller factors and ADP representations¶

Isotropic sites take \(T = \exp(-B_j k^2)\) with \(B_{\mathrm{iso}} = 8\pi^2 U_{\mathrm{iso}}\) (Å²) [Pri04], identical for every image, so it factors out of the orbit sum. Anisotropic sites do not factor:

(4.4)¶\[T_{jm}(\mathbf{h}) \;=\; \exp\!\bigl(-2\pi^2\, \mathbf{q}^\top U^*_j\, \mathbf{q}\bigr), \qquad \mathbf{q} = R_m^\top \mathbf{h}, \qquad U^*_{ij} = U^{ij} a^*_i a^*_j.\]

Source: rietx.crystallography.structure_factor

Three representations of the same tensor appear in the literature, named explicitly here per the IUCr nomenclature report [TBurgiB+96]:

\(U^{ij}\) (Å²) — the CIF _atom_site_aniso_U_ij convention, defined by \(T(\mathbf{h}) = \exp(-2\pi^2 \sum_{ij} U^{ij} h_i h_j a^*_i a^*_j)\); what is stored, so what goes into a CIF is what came out of one.
\(U^*\) (dimensionless) — \(U^*_{ij} = U^{ij} a^*_i a^*_j\), the mean-square displacement tensor in fractional coordinates. \(U^*\) is what transforms as \(U^* \to R\,U^* R^\top\), which makes evaluating the image atom’s factor at \(\mathbf{h}\) identically the parent’s at \(R^\top\mathbf{h}\) — the reciprocal-space action again. This is the form the structure factor uses; working in \(U^*\) makes the identity exact rather than contingent.
\(U_{\mathrm{cart}}\) (Å²) — eigenvalues are the physical mean-square displacements along the ellipsoid axes, and \(U_{\mathrm{eq}} = \operatorname{tr}(U_{\mathrm{cart}})/3\) [FT88].

Positive-definiteness is a property of \(U_{\mathrm{cart}}\), but the three are congruent, so by Sylvester’s law of inertia the signs of the eigenvalues can be tested in any representation. A non-positive-definite tensor raises an ADP_NOT_POSITIVE_DEFINITE diagnostic — the Debye-Waller factor diverges at high \(Q\), so this is not cosmetic — and it is a diagnostic rather than a bound because the constraint couples all six components. Component order throughout is \((U_{11}, U_{22}, U_{33}, U_{12}, U_{13}, U_{23})\). Site-symmetry constraints on both coordinates and ADPs are covered in Parameterisation and constraints.

4.4. Anomalous scattering and the powder average¶

With dispersion on, Friedel’s law dies in a non-centrosymmetric group: \(|F(\mathbf{h})|^2 \ne |F(-\mathbf{h})|^2\). A powder cannot resolve the pair — \(d(\mathbf{h}) = d(-\mathbf{h})\), both land in one peak — so the model must return the orbit average, not one representative’s value. Splitting the species factor into real and imaginary parts,

(4.5)¶\[A(\mathbf{h}) = \sum_j \mathrm{occ}_j\, (f_{0,j} + f'_j) \sum_m T_{jm}\, e^{2\pi i \mathbf{h}\cdot\mathbf{x}_{jm}}, \qquad B(\mathbf{h}) = \sum_j \mathrm{occ}_j\, f''_j \sum_m T_{jm}\, e^{2\pi i \mathbf{h}\cdot\mathbf{x}_{jm}},\]

gives \(F = A + iB\), and since \(T\) is real, \(F(-\mathbf{h}) = \overline{A - iB}\), so

(4.6)¶\[\langle |F|^2 \rangle \;=\; \tfrac{1}{2}\bigl(|F(\mathbf{h})|^2 + |F(-\mathbf{h})|^2\bigr) \;=\; |A|^2 + |B|^2\]

Source: rietx.crystallography.structure_factor

exactly, over the same orbit sums — no second orbit pass and no centro/non-centro case split (in a centrosymmetric group \(A\) and \(B\) share one common phase, so the cross term vanishes identically). \(f'' = 0\) makes \(B \equiv 0\) and recovers \(|F|^2\) bit-identically, so a structure without a dispersion block is unchanged. Merging \(\pm\mathbf{h}\) is therefore exact with or without anomalous scattering — but for two different reasons, and (4.6) is what keeps one representative per Laue orbit the correct thing to enumerate, not an approximation.