# Intensity corrections Each correction in this chapter multiplies the reflection intensity of [(1.2)](forward-model.html.md#equation-fm-rietveld). None of the corrections in this chapter is well judged by $\Delta R_{wp}$. 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). The largest accuracy wins are invisible in it: dispersion on quantitative fractions, absorption on ADPs. Every correction therefore reports what it changed through a record field or a diagnostic, as [Reading a paper against its own numbers](method.html.md#ch-method) describes. ## Lorentz-polarisation $$ \mathrm{Lp}(\theta) \;=\; \frac{K + (1 - K)\cos^2 2\theta}{\sin^2\theta\, \cos\theta}. $$ *Source:* [`rietx.model.corrections.lorentz_polarization`](https://github.com/yue-here/rietx/blob/v1.10.0/src/rietx/model/corrections.py#L10) The $1/(\sin^2\theta \cos\theta)$ Lorentz part is the standard constant-wavelength powder factor (single-crystal rotation Lorentz × powder ring statistics; [[Prince, 2004](references.html.md#id19)] §6.2, [[Klug and Alexander, 1974](references.html.md#id27)]). $K$ is the fraction of the beam polarised perpendicular to the diffraction plane ($\sigma$-polarised). $K = 0.5$ reproduces the unpolarised $(1 + \cos^2 2\theta)/2$, and a synchrotron beam diffracting in the vertical plane has $K \approx 0.99$. A diffracted-beam monochromator sets $K = 1/(1 + \cos^2 2\theta_m)$ [[Azaroff, 1955](references.html.md#id49); [Prince, 2004](references.html.md#id19)]. The familiar 26.6° there is a Cu number and 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 [[McMaster *et al.*, 1969](references.html.md#id62)]: $$ \mu\ [\mathrm{cm}^{-1}] \;=\; \sum_{\mathrm{atoms}} \mathrm{occ}\cdot m \cdot \frac{\sigma_{\mathrm{tot}}\ [\mathrm{barn}]}{V\ [\text{Å}^3]} $$ *Source:* [`rietx.crystallography.attenuation.linear_attenuation`](https://github.com/yue-here/rietx/blob/v1.10.0/src/rietx/crystallography/attenuation.py#L144) The sum runs over the asymmetric-unit sites. A site has occupancy $\mathrm{occ}$ and the multiplicity $m$ of [(9.9)](parameterisation.html.md#equation-par-multiplicity), so $\mathrm{occ}\cdot m$ counts its atoms per cell. $\sigma_{\mathrm{tot}}$ is the species’ total cross section at the working wavelength, and $V$ is the cell volume. One barn is 10⁻²⁴ cm² and one ų is 10⁻²⁴ cm³, so the exponents cancel. Attenuation means beam removal, so the total cross section is the one used, coherent and incoherent scattering included. That is the NIST convention [[Hubbell and Seltzer, 1995](references.html.md#id63)]. The tabulation is a ~2 %-spaced logarithmic grid that cannot represent an absorption edge, so an interval containing an edge raises an error instead of being interpolated. A wavelength that close above an edge also means strong fluorescence, and a refusal is more honest than any number. On a constant-wavelength neutron source the same sum runs over the Sears cross-sections [[Sears, 1992](references.html.md#id20)] instead, since X-ray and neutron attenuation are different quantities and not two precisions of one: $$ \mu\ [\mathrm{cm}^{-1}] \;=\; \sum_{\mathrm{atoms}} \mathrm{occ}\cdot m \cdot \frac{\sigma_{\mathrm{abs}}\,\lambda/\lambda_0 + \sigma_{\mathrm{coh}} + \sigma_{\mathrm{inc}}\ [\mathrm{barn}]} {V\ [\text{Å}^3]}, \qquad \lambda_0 = 1.798\ \text{Å}. $$ *Source:* [`rietx.crystallography.neutron.linear_attenuation_neutron`](https://github.com/yue-here/rietx/blob/v1.10.0/src/rietx/crystallography/neutron.py#L288) $\sigma_{\mathrm{abs}}$ is tabulated for 2200 m/s neutrons, $\lambda_0$, and away from a resonance it follows the $1/v$ law, so it grows linearly in $\lambda$ where an X-ray $\mu/\rho$ rises roughly as $\lambda^{3}$ (falls as $E^{-3}$). The scattering terms are the bound (Sears) values, taken as $\lambda$-independent; for hydrogen the bound value approximates an effective cross-section that in a real solid depends on $\lambda$ and on the proton’s dynamics. In a polycrystal the coherent part is Bragg scattering, and how much of the beam that removes depends on $\lambda$ against the largest $d$-spacing, which is not modelled. Hydrogen is where the two radiations part most: its incoherent 80.26 barn makes it about 93 % of brucite’s neutron $\mu$ and under 1 % of its X-ray one. A resonant absorber (Cd, Sm, Eu, Gd, Yb) is refused. Near a resonance the $1/v$ scaling of the thermal value is wrong in principle. The lowest resonance of each sits at a tabulated energy (`rietx.crystallography.neutron.resonance_wavelengths`), but how near a given $\lambda$ must be to matter is not modelled. That is the neutron twin of the edge refusal above. ## Capillary (cylindrical) absorption The transmission coefficient is the volume average of the attenuation ([[Prince, 2004](references.html.md#id19)] equation 6.3.3.1), $$ A \;=\; \frac{1}{V} \int_V e^{-\mu T}\, dV, $$ *Source:* [`rietx.model.absorption`](https://github.com/yue-here/rietx/blob/v1.10.0/src/rietx/model/absorption.py#L1) with $T$ the total (incident + diffracted) path length. For a cylinder it depends only on $\mu R$ and $\theta$. Rouse et al. [[Rouse *et al.*, 1970](references.html.md#id50)] fit that integral over $0 \le \mu R \le$ 1.0 to better than 0.0035 with $$ A(\mu R, \theta) \;=\; \exp\!\bigl\{ -(a_1 + b_1 \sin^2\theta)\,\mu R - (a_2 + b_2 \sin^2\theta)\,\mu R^2 \bigr\}, $$ $$ a_1 = 1.7133, \quad b_1 = -0.0368, \quad a_2 = -0.0927, \quad b_2 = -0.3750. $$ *Source:* [`rietx.model.absorption`](https://github.com/yue-here/rietx/blob/v1.10.0/src/rietx/model/absorption.py#L1) > **Warning** > > $A$ here is the transmission coefficient, ≤ 1, and the forward model multiplies > it into the intensity. Most tabulations print the absorption correction > $A^* = 1/A \ge 1$ instead, [[Prince, 2004](references.html.md#id19)] Table 6.3.3.2 among them. Getting this > backwards inverts the θ-dependence. An identity test cannot detect the mistake, > since $A(0) = A^*(0) = 1$, and the direction of the θ-dependence can: $A$ > increases with $2\theta$, because the mean path through a cylinder shortens > toward backscatter. One more digit to watch: $b_2 = -0.3750$, against the > $-0.0375$ a scan of the paper prints. [Reading a paper against its own numbers](method.html.md#ch-method) records 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, and $R_{wp}$ cannot move. Its whole physical content is the Biso shift $$ \Delta B \;=\; \frac{c(\mu R)\, \lambda^2}{2}, $$ *Source:* [`rietx.model.absorption.equivalent_delta_biso`](https://github.com/yue-here/rietx/blob/v1.10.0/src/rietx/model/absorption.py#L199) 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. It 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, rather than a merely correlated one. ## Flat-plate absorption The three flat-specimen cases of [[Prince, 2004](references.html.md#id19)] Table 6.3.3.1 follow from the same volume average [(5.4)](#equation-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, because the $\sin\theta$ of the beam footprint cancels against the $\sin\theta$ of the penetration depth. It is identical to the phase scale, and it is what every Bragg-Brentano fit implicitly assumes. The two implemented cases, both normalised: $$ \text{reflection, finite thickness } t:\quad A = 1 - e^{-2\mu t / \sin\theta} \;\longrightarrow\; 1 \text{ as } \mu t \to \infty, $$ *Source:* [`rietx.model.absorption.flat_plate_reflection_absorption`](https://github.com/yue-here/rietx/blob/v1.10.0/src/rietx/model/absorption.py#L216) $$ \text{symmetric transmission}:\quad A = \sec\theta\, e^{-\mu t(\sec\theta - 1)}. $$ *Source:* [`rietx.model.absorption.flat_plate_transmission_absorption`](https://github.com/yue-here/rietx/blob/v1.10.0/src/rietx/model/absorption.py#L232) > **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. > Every other correction here has 0 as its identity. For transmission, $\mu t = 0$ > leaves $\sec\theta$, and that factor is physics: the beam footprint on the tilted > plate grows as $\sec\theta$. The bias directions differ. 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. Neither expression is exactly absorbed by {scale, Biso} as the cylinder’s is: the fraction of $\ln A$ they leave over is 0.2–1.3 % for transmission and a few per cent for finite-thickness reflection, measured at the reflection positions. $\mu t$ is computed from the specimen and never refined, as $\mu R$ is, for the different reason that it is ill-conditioned rather than exactly singular. That leftover 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 that layer at grazing incidence over a long path, and the intensity is depressed. Suortti’s form [[Suortti, 1972](references.html.md#id51)]: $$ R(\theta) \;=\; \frac{a + (1 - a)\, e^{-b/\sin\theta}}{a + (1 - a)\, e^{-b}}, $$ *Source:* [`rietx.model.corrections.surface_roughness_suortti`](https://github.com/yue-here/rietx/blob/v1.10.0/src/rietx/model/corrections.py#L28) normalised so $R(90°) = 1$. Read the two parameters by their physics. $a$ is the intensity fraction surviving at grazing incidence, so $1 - a$ bounds the depression. $b$ is the depleted layer’s dimensionless optical depth, and it sets where in angle the transition falls rather than how deep it goes. For $b \ge 0$ the result is bounded, $0 < R \le 1$, so the correction only ever depresses. The Pitschke et al. form [[Pitschke *et al.*, 1993](references.html.md#id52)]: $$ R(\theta) \;=\; 1 - c\, u (1 - u), \qquad u = \tau / \sin\theta. $$ *Source:* [`rietx.model.corrections.surface_roughness_pitschke`](https://github.com/yue-here/rietx/blob/v1.10.0/src/rietx/model/corrections.py#L62) The paper’s angle-independent term is exactly degenerate with the phase scale, so it is factored out. What remains is the identifiable strength $c$ and the 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 equation 18) it would amplify intensity. The `ROUGHNESS_OUTSIDE_REGIME` diagnostic owns that fence, evaluated at the reflection positions, while the function itself stays smooth and is never clamped, so the Jacobian keeps no kink. Uncorrected roughness biases ADPs severely: Biso refines to −1.9 … −2.5 Ų where the corrected value is +0.3. ## Primary extinction Extinction removes intensity from the strongest reflections, because the diffracted beam re-diffracts inside a coherent domain, a mosaic block. That is primary extinction, and it is the only extinction a powder has: secondary extinction is the attenuation of the beam by other, similarly oriented blocks upstream, and in a powder the corresponding effect is multiple scattering [[Sabine *et al.*, 1988](references.html.md#id55)]. Left uncorrected, the refinement compensates with a spuriously large Biso and a small scale. Sabine’s polycrystalline model [[Sabine, 1985](references.html.md#id53); [Sabine, 1988](references.html.md#id54); [Sabine *et al.*, 1988](references.html.md#id55)] interpolates between the two-beam limits with $\sin^2\theta$ and $\cos^2\theta$ weights: $$ E(hkl) \;=\; E_B \sin^2\theta + E_L \cos^2\theta, \qquad E_B = \frac{1}{\sqrt{1 + x}}, $$ $$ x \;=\; \mathrm{ext} \cdot |F|^2 \cdot \left(\frac{\lambda}{V}\right)^2 \cdot X_{\mathrm{pol}}, \qquad X_{\mathrm{pol}} = 0.079411\cdot\frac{1 + \cos^2 2\theta}{2}, $$ *Source:* [`rietx.model.extinction.sabine_extinction`](https://github.com/yue-here/rietx/blob/v1.10.0/src/rietx/model/extinction.py#L174) with $|F|^2$ entering without multiplicity or Lp. The Laue component is the Laue-case diffracted fraction $\tfrac12[1 - e^{-2\sigma D}]$ integrated over a Lorentzian coupling $\sigma(k)$ and divided by the kinematic intensity [[Prince, 2004](references.html.md#id19)]. That integral has a closed form: $$ E_L \;=\; \frac{1}{\pi x}\int_{-\infty}^{\infty} \tfrac12\left[1 - e^{-2x/(1+u^2)}\right] du \;=\; e^{-x}\left[I_0(x) + I_1(x)\right], \qquad x \ge 0, $$ *Source:* [`rietx.model.extinction._laue_and_deriv`](https://github.com/yue-here/rietx/blob/v1.10.0/src/rietx/model/extinction.py#L124) with $I_0$ and $I_1$ the modified Bessel functions, and $E_L = 1$ at $x < 0$. The papers print two truncated expansions of $E_L$ instead [[Prince, 2004](references.html.md#id19); [Sabine *et al.*, 1988](references.html.md#id55)]: the series $1 - x/2 + x^2/4 - 5x^3/48 + 7x^4/192$ for $x \le 1$ and an asymptote above. The closed form is derived from their stated construction and is printed in neither. It is continuous, with $E_L(1) = 0.67367$, where the printed pair steps by 2.2 % at $x = 1$ and GSAS-II’s six-term series and two-term asymptote step by 3.6 %. $\mathrm{ext}$ is the squared mosaic-block size $D^2$ in µm². The constant 0.079411 is $r_e^2$ in Ų times $10^8$ Ų per µm², to four figures, so $x$ is dimensionless with λ in Å and V in ų. $\mathrm{ext} = 0$ gives $E \equiv \sin^2\theta + \cos^2\theta = 1$ exactly. > **Warning** > > Documented by physics rather than by letter: the Bragg component weights > $\sin^2\theta$ and the Laue component $\cos^2\theta$. That is the opposite of > the naive reading, because backscattering ($2\theta \to 180°$) is the Bragg-case > limit and forward scattering the Laue-case limit. Real powder data reach $x > 1$. On the [QPA guide](../guides/qpa.html.md)’s first fit, CaF₂’s reflections run to $x = 6.5$ at the answer. Until 1.10, rietx used GSAS-II’s stepped pair, and that fit stopped with one reflection on the step. CaF₂’s esd then depended on whether the solver’s last Jacobian straddled it: 0.21 wt % on one platform and 2.24 on another. ## Preferred orientation (March-Dollase) A non-random crystallite orientation distribution biases intensities. The March distribution [[March, 1932](references.html.md#id56)] as folded into Rietveld refinement by Dollase [[Dollase, 1986](references.html.md#id57)] is a per-reflection multiplier averaged over the symmetry orbit (multiplicity $M$): $$ P_{hkl} \;=\; \frac{1}{M} \sum_{m \in \mathrm{orbit}} \left[ r^2 \cos^2\alpha_m + \frac{\sin^2\alpha_m}{r} \right]^{-3/2}, $$ *Source:* [`rietx.model.preferred_orientation.march_dollase_factors`](https://github.com/yue-here/rietx/blob/v1.10.0/src/rietx/model/preferred_orientation.py#L142) 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: $$ \cos\alpha \;=\; \frac{\mathbf{h}_m \cdot G^* \cdot \mathbf{a}} {\sqrt{(\mathbf{h}_m \cdot G^* \cdot \mathbf{h}_m) (\mathbf{a} \cdot G^* \cdot \mathbf{a})}}. $$ *Source:* [`rietx.model.preferred_orientation.cos2_alpha`](https://github.com/yue-here/rietx/blob/v1.10.0/src/rietx/model/preferred_orientation.py#L79) At $r = 1$ every bracket is 1, so $P \equiv 1$ exactly. Off is the identity, for every reflection and every cell. Friedel mates give identical brackets, so orbit merging is unaffected. > **Warning** > > Codes disagree on the sign convention of $r$, so the physics is the contract > and the letter is not. For a reflection parallel to the axis, $P = r^{-3}$, so > $r < 1$ enhances axial reflections. In Bragg-Brentano reflection geometry with > axis = plate normal, $r < 1$ means a platy habit and $r > 1$ an acicular one. In > transmission (capillary) geometry the sense reverses for the same axis choice. > The correction is geometry-agnostic. The interpretation of $r$ is not. ## Preferred orientation (spherical harmonics) March-Dollase describes one fibre axis with one number. A texture with no single axis needs the general description, a series of symmetrised spherical harmonics [[Von Dreele, 1997](references.html.md#id5)]. Under cylindrical sample symmetry the multiplier on a reflection’s intensity is $$ \begin{gathered} P_{h} \;=\; 1 + \sum_{l = 2, 4, \ldots}^{L} \frac{4\pi}{2l+1} \sum_{n} C_l^{\,n}\, k_l(\psi)\, K_l^{\,n}(\mathbf{h}), \\ k_l(\psi) = \sqrt{\frac{2l+1}{4\pi}}\, P_l(\cos\psi). \end{gathered} $$ *Source:* [`rietx.crystallography.spherical_harmonics.texture_basis`](https://github.com/yue-here/rietx/blob/v1.10.0/src/rietx/crystallography/spherical_harmonics.py#L449) $L$ is the declared order. Only even $l$ appear, because a powder pattern obeys Friedel’s law, and $l = 0$ is the phase scale. Each $K_l^n$ is a real harmonic of the reflection’s direction, orthonormal over the sphere and invariant under the Laue group. The harmonics follow Bunge [[Bunge, 1982](references.html.md#id6)], with no Condon-Shortley phase. The crystal frame puts $z$ on the principal axis and $x$ on the next cell edge [[Popa, 1992](references.html.md#id7)]. Which $(l, n)$ the class allows is read off the operators. Every $C = 0$ gives $P \equiv 1$ exactly. $\psi$ is the angle between the scattering vector and the specimen’s symmetry axis. In symmetric reflection the scattering vector lies along the plate normal, so $\psi = 0$. In symmetric transmission and in a spinning capillary it is perpendicular to the axis, so $\psi = 90°$ [[Järvinen, 1993](references.html.md#id8)]. The size of the texture is the texture index $J = 1 + \sum C^2/(2l+1)$, which is 1 for a random powder. > **Warning** > > GSAS-II puts the Bragg-Brentano $\psi$ at 90°. For the same pattern, this > package’s coefficient is GSAS-II’s times $P_l(0)$, which is $-1/2$ at > $l = 2$. In the five Laue classes without mirrors ($\bar 1$, $2/m$, $4/m$, > $\bar 3$, $6/m$), GSAS-II’s $n \ne 0$ functions also have norm $\sqrt 2$, so > those coefficients take a further factor $\sqrt 2$. > > GSAS-II 5.8.4’s functions are not invariant under the Laue group in three > places: the $n \ne 0$ terms of $\bar 1$ and $2/m$, every term on rhombohedral > axes, and the $n = 3$ terms of $\bar 3 1m$. Each takes two values on one set of > equivalent reflections, so those coefficients have no counterpart here. GSAS-II > also gives $m\bar 3$ one function fewer at $l = 6$ than the group allows. ## Quantitative phase analysis and microabsorption Weight fractions follow the Hill-Howard scale-factor relation [[Hill and Howard, 1987](references.html.md#id59)] (see also [[Bish and Howard, 1988](references.html.md#id60)]): $$ W_p \;=\; \frac{S_p\, (Z M V)_p}{\sum_q S_q\, (Z M V)_q}, $$ *Source:* [`rietx.optimize.qpa.weight_fractions`](https://github.com/yue-here/rietx/blob/v1.10.0/src/rietx/optimize/qpa.py#L170) 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 $ZM = \sum \mathrm{occ}\cdot m\cdot A$, and the $Z/M$ split is display-only. Here $m$ is the site multiplicity and $A$ the atomic weight: the IUPAC standard weight for an element, and for a nuclide (`D`, `2H`, `7Li`) its own mass, 2.0141 for ²H and the mass number otherwise, within 0.26 % of the nuclide’s mass above A = 4. These are fractions of the modelled crystalline content, so they still sum to 1 when an amorphous fraction is present. When phases differ in absorption, coarse particles of an absorbing phase shadow their own interiors. That is the Brindley microabsorption effect [[Brindley, 1945](references.html.md#id58)]. In the parallel-path approximation for a sphere of radius $R$: $$ \tau(x) \;=\; \frac{3\,[\,2 - e^{-u}(u^2 + 2u + 2)\,]}{u^3}, \qquad u = 2x, \quad x = (\mu_p - \bar\mu)\, R, $$ *Source:* [`rietx.optimize.qpa.brindley_tau`](https://github.com/yue-here/rietx/blob/v1.10.0/src/rietx/optimize/qpa.py#L211) 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 [[Taylor and Matulis, 1991](references.html.md#id61)], 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 and recorded mistake. The corrected fractions are reported alongside the uncorrected Hill-Howard numbers, and never silently substituted.