5. Intensity corrections¶

Each correction in this chapter multiplies the reflection intensity of (1.2). A method result from shipping all of them, recorded once here rather than per section: not one is well judged by ΔRwp — one provably cannot move it (capillary absorption, an exact reparameterisation), one moves it the wrong way when it is right (a declared flat-plate thickness on a thick specimen), and the largest accuracy wins (dispersion on quantitative fractions, absorption on ADPs) are invisible in it. Every correction therefore reports what it changed through a record field or a diagnostic; see Reading a paper against its own numbers.

5.1. Lorentz-polarisation¶

(5.1)¶\[\mathrm{Lp}(\theta) \;=\; \frac{K + (1 - K)\cos^2 2\theta}{\sin^2\theta\, \cos\theta}.\]

Source: rietx.model.corrections.lorentz_polarization

The \(1/(\sin^2\theta \cos\theta)\) Lorentz part is the standard constant-wavelength powder factor (single-crystal rotation Lorentz × powder ring statistics; [Pri04] §6.2, [KA74]). \(K\) is the fraction of the beam polarised perpendicular to the diffraction plane (\(\sigma\)-polarised): \(K = 0.5\) reproduces the unpolarised \((1 + \cos^2 2\theta)/2\); a synchrotron beam diffracting in the vertical plane has \(K \approx 0.99\). A diffracted-beam monochromator sets \(K = 1/(1 + \cos^2 2\theta_m)\) [Aza55, Pri04] — and the familiar 26.6° there is a Cu number, not a property of the graphite crystal.

5.2. Attenuation coefficients¶

Specimen absorption needs \(\mu\), computed from the refined cell contents and the McMaster total cross sections [MDGMH69]:

(5.2)¶\[\mu\ [\mathrm{cm}^{-1}] \;=\; \sum_{\mathrm{atoms}} \mathrm{occ}\cdot m \cdot \frac{\sigma_{\mathrm{tot}}\ [\mathrm{barn}]}{V\ [\text{Å}^3]}\]

Source: rietx.crystallography.attenuation

(1 barn = 10⁻²⁴ cm² and 1 Å³ = 10⁻²⁴ cm³, so the exponents cancel). Attenuation means beam removal, so the total cross section including coherent and incoherent scattering is used — the NIST convention [HS95]. The tabulation is a ~2 %-spaced logarithmic grid that cannot represent an absorption edge, so an interval containing an edge is refused with an error rather than interpolated: a wavelength that close above an edge also means strong fluorescence, and a refusal is more honest than any number.

5.3. Capillary (cylindrical) absorption¶

The transmission coefficient is the volume average of the attenuation ([Pri04] eq. 6.3.3.1),

(5.3)¶\[A \;=\; \frac{1}{V} \int_V e^{-\mu T}\, dV,\]

Source: rietx.model.absorption

with \(T\) the total (incident + diffracted) path length. For a cylinder it depends only on \(\mu R\) and \(\theta\); Rouse et al. [RCYC70] fit that integral over \(0 \le \mu R \le\) 1.0 to better than 0.0035 with

(5.4)¶\[A(\mu R, \theta) \;=\; \exp\!\bigl\{ -(a_1 + b_1 \sin^2\theta)\,\mu R - (a_2 + b_2 \sin^2\theta)\,\mu R^2 \bigr\},\]

(5.5)¶\[a_1 = 1.7133, \quad b_1 = -0.0368, \quad a_2 = -0.0927, \quad b_2 = -0.3750.\]

Source: rietx.model.absorption

Warning

\(A\) here is the transmission coefficient, ≤ 1, which the forward model multiplies into the intensity. Most tabulations — including [Pri04] Table 6.3.3.2 — print the absorption correction \(A^* = 1/A \ge 1\) instead. Getting this backwards inverts the θ-dependence, and \(A(0) = A^*(0) = 1\) means an identity test cannot detect it: the direction of the θ-dependence is what does (\(A\) increases with \(2\theta\), because the mean path through a cylinder shortens toward backscatter). And \(b_2 = -0.3750\), not the \(-0.0375\) a scan of the paper prints — see Reading a paper against its own numbers for how that was settled.

The expression factors exactly into \(A = K(\mu R)\cdot\exp(+c(\mu R) \sin^2\theta)\) — a constant times a Debye-Waller shape. Applying it to a model with free scale and displacement parameters is therefore an exact reparameterisation: Rwp cannot move. Its entire physical content is the Biso shift

(5.6)¶\[\Delta B \;=\; \frac{c(\mu R)\, \lambda^2}{2},\]

Source: rietx.model.absorption.equivalent_delta_biso

0.13 Å² at \(\mu R = 0.5\) and 0.49 Å² at \(\mu R = 1.0\) for Cu Kα — neglecting capillary absorption biases Biso low by that much. This is also why \(\mu R\) is a plain float and never refinable: a free \(\mu R\) is an exactly singular direction in the normal equations, not merely a correlated one.

5.4. Flat-plate absorption¶

The three flat-specimen cases of [Pri04] Table 6.3.3.1 follow from the same volume average (5.3), each in closed form — nothing is a fit, so there are no coefficients to transcribe wrongly. The thick reflection specimen, case (1a), gives \(A = 1/2\mu\) with no θ-dependence at all (the \(\sin\theta\) of the beam footprint cancels against the \(\sin\theta\) of the penetration depth); it is identical to the phase scale and is what every Bragg-Brentano fit implicitly assumes. The two implemented cases, both normalised:

(5.7)¶\[\text{reflection, finite thickness } t:\quad A = 1 - e^{-2\mu t / \sin\theta} \;\longrightarrow\; 1 \text{ as } \mu t \to \infty,\]

Source: rietx.model.absorption.flat_plate_reflection_absorption

(5.8)¶\[\text{symmetric transmission}:\quad A = \sec\theta\, e^{-\mu t(\sec\theta - 1)}.\]

Source: rietx.model.absorption.flat_plate_transmission_absorption

Warning

The two cases take opposite answers about what “off” means. For reflection the identity is an infinitely thick specimen — \(\mu t\) absent means thick, and \(\mu t = 0\) is a specimen of no thickness, which diffracts nothing and raises. This is the reverse of every other correction here, where 0 is the identity. For transmission, \(\mu t = 0\) leaves \(\sec\theta\) — physics, not a leftover: the beam footprint on the tilted plate grows as \(\sec\theta\).

Bias directions: finite-thickness reflection depresses high-angle intensity, so a Biso refined without it comes back too large — the opposite sign to the capillary. Transmission flips sign with thickness. Unlike the cylinder, neither expression is exactly absorbed by {scale, Biso}: the unabsorbed fraction of \(\ln A\) is 0.2–1.3 % for transmission and a few per cent for finite-thickness reflection, measured at the reflection positions. \(\mu t\), like \(\mu R\), is computed from the specimen and never refined — but for a different reason (ill-conditioned, not exactly singular), and the unabsorbed fraction is reported so a caller can disagree.

5.5. Surface roughness¶

A rough or loosely packed flat specimen has a packing-density deficit in its top layer; at low θ the beam crosses it at grazing incidence over a long path, depressing intensity. Suortti’s form [Suo72]:

(5.9)¶\[R(\theta) \;=\; \frac{a + (1 - a)\, e^{-b/\sin\theta}}{a + (1 - a)\, e^{-b}},\]

Source: rietx.model.corrections.surface_roughness_suortti

normalised so \(R(90°) = 1\). Physics, not letters: \(a\) is the intensity fraction surviving at grazing incidence, so \(1 - a\) bounds the depression; \(b\) is the depleted layer’s dimensionless optical depth, which sets where in angle the transition falls — not how deep it goes. For \(b \ge 0\) the result is bounded \(0 < R \le 1\): the correction only ever depresses.

The Pitschke et al. form [PHM93]:

(5.10)¶\[R(\theta) \;=\; 1 - c\, u (1 - u), \qquad u = \tau / \sin\theta.\]

Source: rietx.model.corrections.surface_roughness_pitschke

The paper’s angle-independent term is factored out because it is exactly degenerate with the phase scale, leaving the identifiable strength \(c\) and roughness parameter τ. This model has a validity range and does not police it: \(R\) is monotone in θ only while \(\sin\theta \ge 2\tau\), and beyond \(\sin\theta = \tau\) (its Eq 18) it would amplify intensity. The ROUGHNESS_OUTSIDE_REGIME diagnostic owns the fence — evaluated at the reflection positions, not over the 2θ grid — while the function itself stays smooth and unclamped so the Jacobian keeps no kink. Left uncorrected, roughness biases ADPs severely (the classic result: Biso refining to −1.9 … −2.5 Å² where the corrected value is +0.3).

5.6. Secondary extinction¶

Extinction removes intensity from the strongest reflections because the diffracted beam re-diffracts inside a coherent domain; uncorrected, the refinement compensates with a spuriously large Biso and small scale. Sabine’s polycrystalline model [Sab85, Sab88, SVDJorgensen88] blends the two-beam limits by the fraction of a random powder in each geometry:

(5.11)¶\[E(hkl) \;=\; E_B \sin^2\theta + E_L \cos^2\theta, \qquad E_B = \frac{1}{\sqrt{1 + x}},\]

(5.12)¶\[x \;=\; \mathrm{ext} \cdot |F|^2 \cdot \left(\frac{\lambda}{V}\right)^2 \cdot X_{\mathrm{pol}}, \qquad X_{\mathrm{pol}} = 0.079411\cdot\frac{1 + \cos^2 2\theta}{2},\]

Source: rietx.model.extinction

with \(E_L\) a six-term series in \(x\) for \(0 < x \le 1\) and a two-term asymptote above, \(E_L = 1\) at \(x \le 0\), and \(|F|^2\) entering without multiplicity or Lp. \(\mathrm{ext} = 0\) gives \(E \equiv \sin^2\theta + \cos^2\theta = 1\) exactly.

Warning

Documented by physics, not letter: the Bragg component weights \(\sin^2\theta\) and the Laue component \(\cos^2\theta\) — the opposite of the naive reading, because backscattering (\(2\theta \to 180°\)) is the Bragg-case limit and forward scattering the Laue-case limit. The two Laue branches deliberately do not join continuously at \(x = 1\) (a ~2 % step, inherited verbatim from the cross-code reference and out of reach for real powder data, where \(x \ll 1\)); smoothing it would break the cross-code golden.

5.7. Preferred orientation (March-Dollase)¶

A non-random crystallite orientation distribution biases intensities. The March distribution [Mar32] as folded into Rietveld refinement by Dollase [Dol86] is a per-reflection multiplier averaged over the symmetry orbit (multiplicity \(M\)):

(5.13)¶\[P_{hkl} \;=\; \frac{1}{M} \sum_{m \in \mathrm{orbit}} \left[ r^2 \cos^2\alpha_m + \frac{\sin^2\alpha_m}{r} \right]^{-3/2},\]

Source: rietx.model.preferred_orientation

where \(\alpha_m\) is the angle between the preferred-orientation axis and the scattering vector of equivalent \(m\). Both are reciprocal-lattice directions (integer \(hkl\)), so the angle uses the reciprocal metric:

(5.14)¶\[\cos\alpha \;=\; \frac{\mathbf{h}_m \cdot G^* \cdot \mathbf{a}} {\sqrt{(\mathbf{h}_m \cdot G^* \cdot \mathbf{h}_m) (\mathbf{a} \cdot G^* \cdot \mathbf{a})}}.\]

Source: rietx.model.preferred_orientation

At \(r = 1\) every bracket is 1, so \(P \equiv 1\) exactly — the identity when off, for every reflection and cell. Friedel mates give identical brackets, so orbit merging is unaffected.

Warning

Codes disagree on the sign convention of \(r\), so the physics — not the letter — is the contract. For a reflection parallel to the axis, \(P = r^{-3}\): \(r < 1\) enhances axial reflections. In Bragg-Brentano reflection geometry with axis = plate normal, \(r < 1\) ⇒ platy habit and \(r > 1\) ⇒ acicular; in transmission (capillary) geometry the sense reverses for the same axis choice. The correction itself is geometry-agnostic; the interpretation of r is not.

Spherical-harmonics texture [VD97] is not implemented; it is planned for v2.

5.8. Quantitative phase analysis and microabsorption¶

Weight fractions follow the Hill-Howard scale-factor relation [HH87] (see also [BH88]):

(5.15)¶\[W_p \;=\; \frac{S_p\, (Z M V)_p}{\sum_q S_q\, (Z M V)_q},\]

Source: rietx.optimize.qpa

with \(Z\) formula units per cell, \(M\) the formula mass and \(V\) the cell volume — all derived from the refined model. Occupancies enter the mass, so the load-bearing quantity is the cell mass \(ZM = \sum \mathrm{occ}\cdot m\cdot A\); the \(Z/M\) split is display-only. These are fractions of the modelled crystalline content: an amorphous fraction still makes them sum to 1.

When phases differ in absorption, coarse particles of an absorbing phase shadow their own interiors — the Brindley microabsorption effect [Bri45]. In the parallel-path approximation for a sphere of radius \(R\):

(5.16)¶\[\tau(x) \;=\; \frac{3\,[\,2 - e^{-u}(u^2 + 2u + 2)\,]}{u^3}, \qquad u = 2x, \quad x = (\mu_p - \bar\mu)\, R,\]

Source: rietx.optimize.qpa.brindley_tau

exact at \(\tau(0) = 1\) and \(\tau > 1\) for a phase less absorbing than the matrix. Inside the validity domain this agrees to <1 % with Brindley’s own geometry-averaged table (as represented by the two independently published fits used by FullProf and MAUD [TM91], which themselves scatter by ~1 %). The validity fence is \(\mu R \le\) 0.05 — derived from Brindley’s \(\mu D \le 0.1\) with \(D = 2R\) the particle diameter; conflating the two conventions is a real, recorded mistake. The corrected fractions are reported alongside the uncorrected Hill-Howard numbers, never silently substituted.