(ch-corrections)= # Intensity corrections Each correction in this chapter multiplies the reflection intensity of {eq}`fm-rietveld`. 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 {ref}`ch-method`. ## Lorentz-polarisation ```{math} :label: corr-lp \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; {cite}`itc-c` §6.2, {cite}`klug1974`). $K$ 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 $K = 1/(1 + \cos^2 2\theta_m)$ {cite}`itc-c,azaroff1955` — and the familiar 26.6° there is a *Cu* number, not a property of the graphite crystal. ## Attenuation coefficients Specimen absorption needs $\mu$, computed from the refined cell contents and the McMaster total cross sections {cite}`mcmaster1969`: ```{math} :label: corr-mu \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 ų = 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 {cite}`hubbell1995`. 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. ## Capillary (cylindrical) absorption The transmission coefficient is the volume average of the attenuation ({cite}`itc-c` eq. 6.3.3.1), ```{math} :label: corr-itc 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. {cite}`rouse1970` fit that integral over $0 \le \mu R \le$ {{ CYLINDER_MU_R_MAX }} to better than 0.0035 with ```{math} :label: corr-rouse 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\}, ``` ```{math} :label: corr-rouse-coeff 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 {cite}`itc-c` 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 {ref}`ch-method` 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 ```{math} :label: corr-deltab \Delta B \;=\; \frac{c(\mu R)\, \lambda^2}{2}, ``` *Source:* `rietx.model.absorption.equivalent_delta_biso` 0.13 Ų at $\mu R = 0.5$ and 0.49 Ų 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. ## Flat-plate absorption The three flat-specimen cases of {cite}`itc-c` Table 6.3.3.1 follow from the same volume average {eq}`corr-itc`, 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: ```{math} :label: corr-fp2 \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` ```{math} :label: corr-fp3a \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. ## 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 {cite}`suortti1972`: ```{math} :label: corr-suortti 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 {cite}`pitschke1993`: ```{math} :label: corr-pitschke 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 Ų where the corrected value is +0.3). ## 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 {cite}`sabine1985,sabine1988,sabine1988b` blends the two-beam limits by the fraction of a random powder in each geometry: ```{math} :label: corr-sabine E(hkl) \;=\; E_B \sin^2\theta + E_L \cos^2\theta, \qquad E_B = \frac{1}{\sqrt{1 + x}}, ``` ```{math} :label: corr-sabine-x 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. ``` ## Preferred orientation (March-Dollase) A non-random crystallite orientation distribution biases intensities. The March distribution {cite}`march1932` as folded into Rietveld refinement by Dollase {cite}`dollase1986` is a per-reflection multiplier averaged over the symmetry orbit (multiplicity $M$): ```{math} :label: corr-md 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: ```{math} :label: corr-md-angle \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 {cite}`vondreele1997` is not implemented; it is planned for v2. ## Quantitative phase analysis and microabsorption Weight fractions follow the Hill-Howard scale-factor relation {cite}`hill1987` (see also {cite}`bish1988`): ```{math} :label: corr-qpa 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 {cite}`brindley1945`. In the parallel-path approximation for a sphere of radius $R$: ```{math} :label: corr-brindley \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 {cite}`taylor1991`, which themselves scatter by ~1 %). The validity fence is $\mu R \le$ {{ BRINDLEY_MU_R_FENCE }} — 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.