5. Intensity corrections#

Each correction in this chapter multiplies the reflection intensity of (1.2). 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 describes.

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; [Prince, 2004] §6.2, [Klug and Alexander, 1974]). \(K\) 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; Prince, 2004]. The familiar 26.6° there is a Cu number and 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 [McMaster et al., 1969]:

(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.linear_attenuation

The sum runs over the asymmetric-unit sites. A site has occupancy \(\mathrm{occ}\) and the multiplicity \(m\) of (9.9), 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 Å³ 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]. 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] instead, since X-ray and neutron attenuation are different quantities and not two precisions of one:

(5.3)#\[\mu\ [\mathrm{cm}^{-1}] \;=\; \sum_{\mathrm{atoms}} \mathrm{occ}\cdot m \cdot \frac{\sigma_{\mathrm{abs}}\,\lambda/\lambda_0 + \sigma_{\mathrm{coh}} + \sigma_{\mathrm{inc}}\ [\mathrm{barn}]} {V\ [\text{Å}^3]}, \qquad \lambda_0 = 1.798\ \text{Å}.\]

Source: rietx.crystallography.neutron.linear_attenuation_neutron

\(\sigma_{\mathrm{abs}}\) 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.

5.3. Capillary (cylindrical) absorption#

The transmission coefficient is the volume average of the attenuation ([Prince, 2004] equation 6.3.3.1),

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

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. [Rouse et al., 1970] fit that integral over \(0 \le \mu R \le\) 1.0 to better than 0.0035 with

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

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

Source: rietx.model.absorption

Warning

\(A\) here is the transmission coefficient, ≤ 1, and the forward model multiplies it into the intensity. Most tabulations print the absorption correction \(A^* = 1/A \ge 1\) instead, [Prince, 2004] 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 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

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

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. 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.

5.4. Flat-plate absorption#

The three flat-specimen cases of [Prince, 2004] Table 6.3.3.1 follow from the same volume average (5.4), 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:

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

Source: rietx.model.absorption.flat_plate_reflection_absorption

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

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. 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.

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 that layer at grazing incidence over a long path, and the intensity is depressed. Suortti’s form [Suortti, 1972]:

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

Source: rietx.model.corrections.surface_roughness_suortti

normalised so \(R(90°) = 1\). 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]:

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

Source: rietx.model.corrections.surface_roughness_pitschke

The paper’s angle-independent term is 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 Å² where the corrected value is +0.3.

5.6. 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]. Left uncorrected, the refinement compensates with a spuriously large Biso and a small scale. Sabine’s polycrystalline model [Sabine, 1985; Sabine, 1988; Sabine et al., 1988] interpolates between the two-beam limits with \(\sin^2\theta\) and \(\cos^2\theta\) weights:

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

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

Source: rietx.model.extinction.sabine_extinction

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]. That integral has a closed form:

(5.14)#\[E_L \;=\; \frac{1}{\pi x}\int_{-\infty}^{\infty} \tfrac12\left[1 - e^{-2x/(1+u^2)}\right] du \;=\; e^{-x}\left[I_0(x) + I_1(x)\right], \qquad x \ge 0,\]

Source: rietx.model.extinction._laue_and_deriv

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; Sabine et al., 1988]: 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 Å² times \(10^8\) Å² per µm², to four figures, so \(x\) is dimensionless with λ in Å and V in Å³. \(\mathrm{ext} = 0\) 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’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.

5.7. Preferred orientation (March-Dollase)#

A non-random crystallite orientation distribution biases intensities. The March distribution [March, 1932] as folded into Rietveld refinement by Dollase [Dollase, 1986] is a per-reflection multiplier averaged over the symmetry orbit (multiplicity \(M\)):

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

Source: rietx.model.preferred_orientation.march_dollase_factors

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:

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

Source: rietx.model.preferred_orientation.cos2_alpha

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.

5.8. 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]. Under cylindrical sample symmetry the multiplier on a reflection’s intensity is

(5.17)#\[\begin{split}\begin{gathered} P_{h} \;=\; 1 + \sum_{l = 2, 4, \ldots}^{L} \frac{4\pi}{2l+1} \sum_{n} C_l^{\,n}\, k_l(\psi)\, K_l^{\,n}(\mathbf{h}), \\ k_l(\psi) = \sqrt{\frac{2l+1}{4\pi}}\, P_l(\cos\psi). \end{gathered}\end{split}\]

Source: rietx.crystallography.spherical_harmonics.texture_basis

\(L\) 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], with no Condon-Shortley phase. The crystal frame puts \(z\) on the principal axis and \(x\) on the next cell edge [Popa, 1992]. 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]. 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.

5.9. Quantitative phase analysis and microabsorption#

Weight fractions follow the Hill-Howard scale-factor relation [Hill and Howard, 1987] (see also [Bish and Howard, 1988]):

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

Source: rietx.optimize.qpa.weight_fractions

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]. In the parallel-path approximation for a sphere of radius \(R\):

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

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 [Taylor and Matulis, 1991], 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.