6. Microstructure

6.1. Isotropic size and strain

Sample broadening rides on the instrument profile through the width laws of (3.1) and (3.2): crystallite-size broadening varies as \(1/\cos\theta\) (Scherrer; Lorentzian \(X_s\), Gaussian variance \(P/\cos^2\theta\)), microstrain broadening as \(\tan\theta\) (Lorentzian \(Y_s\), Gaussian variance \(U_s \tan^2\theta\)). The instrument ⊕ sample split is a workflow, not just an equation: calibrate \(U, V, W, X, Y\) on a line-width standard with its certified cell held fixed, freeze them, and refine only the sample terms on the specimen.

These laws depend on \(hkl\) only through θ. Real strained powders break that — \((00l)\) and \((hk0)\) can differ threefold at the same \(2\theta\).

6.2. Stephens anisotropic strain

Stephens’ phenomenological model [Ste99] lets every crystallite carry its own lattice metric. With

(6.1)\[M_{hkl} \;\equiv\; \frac{1}{d^2_{hkl}} \;=\; \mathbf{h}\cdot G^*\cdot \mathbf{h}^\top \quad [\text{Å}^{-2}],\]

the spread of a quadratic form’s coefficients makes the variance of \(M\) a homogeneous quartic in \((h, k, l)\):

(6.2)\[\sigma^2(M) \;=\; 10^{-12} \sum_{H+K+L=4} S_{HKL}\; h^H k^K l^L,\]

Source: rietx.crystallography.stephens

fifteen monomials, hence at most fifteen coefficients. Since \(2\theta = 2\arcsin(\lambda\sqrt{M}/2)\) gives \(d(2\theta)/dM = \tan\theta / M\), the contribution in the deg-2θ FWHM units the Lorentzian strain term already uses is

(6.3)\[\Lambda(hkl) \;=\; \frac{180}{\pi}\cdot 10^{-6}\cdot d^2_{hkl} \cdot \sqrt{\textstyle\sum_{HKL} S_{HKL}\, h^H k^K l^L} \quad [\deg],\]

Source: rietx.crystallography.stephens

added to the Lorentzian FWHM as \(\Lambda(hkl)\cdot\tan\theta\) — the first width in the model that depends on \(hkl\) rather than only on θ.

Warning

Three independent labelling conventions sit behind these \(S_{HKL}\), and getting any one wrong rescales every published number. A manual that reproduced Stephens’ equation (1) without them would be worse than none — a reader would transfer literature values straight in and get a wrong width law that still refines.

  1. \(\sqrt{\sum S\cdot\text{monomial}}\cdot d^2\cdot 10^{-6}\) is the FWHM of the \(\Delta M/M = 2\Delta d/d\) distribution, not its standard deviation — no \(\sqrt{8\ln 2}\) appears anywhere.

  2. The coefficients are carried in 10⁻¹² Å⁻⁴, not physical Å⁻⁴ — and that is load-bearing numerically, not cosmetic: the shared finite-difference step is absolute below 1, so a coefficient at its physical ~10⁻⁸ Å⁻⁴ magnitude would be differenced with a step 100× its own value.

  3. They multiply the literal monomials \(h^H k^K l^L\). Other codes fold symmetry multiplicities into their templates (writing the cubic S₂₂₀ term as \(3(h^2k^2 + h^2l^2 + k^2l^2)\), say), so their printed values differ by small integer factors as well. Never transfer a literature \(S_{HKL}\) without checking numerically.

6.3. Symmetry and the allowed coefficients

\(\sigma^2(M)\) must be invariant under the Laue group. Miller indices transform under the reciprocal-space action \(\mathbf{h}' = R^\top\mathbf{h}\), which induces a 15×15 integer action \(A(R)\) on the monomial coefficients; the allowed \(S_{HKL}\) span \(\bigcap_R \ker(A(R) - I)\), computed as an exact rational nullspace — the rank-4 twin of the rank-2 construction used for ADPs [PP66], sharing the same kernel. Degree 4 is inversion-even, so no Laue classification is needed. The derived dimensions reproduce Stephens’ Table 1: \(m\bar 3m\) 2, \(6/mmm\) and \(6/m\) 3, \(\bar 3m1\) and \(\bar 31m\) 4, \(\bar 3\) 5, \(4/mmm\) 4, \(4/m\) 5, \(mmm\) 6, \(2/m\) 9, \(\bar 1\) 15.

The coefficients refine as absolute degrees of freedom on that basis — a set outside the allowed subspace raises rather than being symmetrised. A Stephens block locks the scalar lor_strain: its isotropic direction is identically that column, so the block subsumes it, and it must be freed in the sample-broadening stage, not after.

6.4. The positivity cone, the seed, and how to read the guard

\(\sigma^2(M) \ge 0\) for every \(hkl\) is a cone coupling all fifteen coefficients — it cannot be a box bound. Under the default TRF driver it is a guard (STEPHENS_STRAIN_NOT_POSITIVE); under the bounded-LM driver it is carried as a linear inequality and the guard falls silent because there is nothing left to report (Estimation). Read a firing as “these coefficients are not quotable”, never as evidence of anisotropy — and note that zero is on the cone, not outside it: the guard’s test is one-sided, and an earlier ≤ 0 form that flagged the inert all-zero block produced a since-withdrawn claim about isotropic specimens.

The isotropic limit \(S = \varepsilon^2\,[M^2]\) lies exactly in the allowed subspace for every symmetry and is both the seed and the only legal start: at \(S \equiv 0\) the square root in (6.3) has unbounded slope, so strain stages seed through a dedicated mechanism rather than a generic parameter seed.