6. Microstructure¶
6.1. Isotropic size and strain¶
Sample broadening rides on the instrument profile through the width laws of (3.1) and (3.3). Crystallite-size broadening varies as \(1/\cos\theta\) (Scherrer; Lorentzian \(X_s\), Gaussian variance \(P/\cos^2\theta\)), and microstrain broadening as \(\tan\theta\) (Lorentzian \(Y_s\), Gaussian variance \(U_s \tan^2\theta\)). The instrument ⊕ sample split is a workflow as much as 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. Reading a width as a size¶
A width in degrees does not transfer between instruments. The crystallite size it implies does. Scherrer’s relation [Scherrer, 1918] reads a single line’s breadth as an apparent size,
Source: rietx.model.profiles.caglioti.apparent_size
with \(2w\) the FWHM in radians of \(2\theta\), which is Langford and Wilson’s symbol for it [Langford and Wilson, 1978]. Their \(\beta\) is the integral breadth, the peak area divided by the peak height, and the notation table of rietx manual rules that out as a width measure here. The lower-case \(2w\) is a width in radians and has nothing to do with the Caglioti \(W\) of (3.1), which is a variance in deg². \(K\) depends on the crystallite shape and on which of the two breadths is used: 0.89 for the FWHM of a sphere against 1.0747 for its integral breadth. An apparent size is therefore an order-of-magnitude statement rather than a quotable two-figure one. It is also a lower bound unless the instrument and strain contributions have been taken out first, since Scherrer attributes every degree it is handed to size.
The size transfers and the degrees do not, for the same reason that working in \(Q\) helps. With \(Q = 4\pi\sin\theta/\lambda\) the local Jacobian is \(dQ/d(2\theta) = 2\pi\cos\theta/\lambda\), and the \(\cos\theta\) of (6.1) cancels against it:
Source: rietx.model.profiles.caglioti.delta_q_fwhm
independent of both \(\lambda\) and \(\theta\), so Scherrer broadening is constant in Q. The size law of (3.3) is itself \(1/\cos\theta\), so the same cancellation removes the reference angle entirely, and a size coefficient maps to one size for the whole pattern,
Source: rietx.model.profiles.caglioti.apparent_size_from_size_coefficient
reading \(\sqrt{P}\) in place of \(X_s\) for the Gaussian variance coefficient of (3.1). The right-hand form is the seeding direction. It gives a width for a specimen whose size is already known from a micrograph, a synthesis or an earlier refinement, in place of a default calibrated on a synchrotron linewidth.
Warning
Only the \(1/\cos\theta\) coefficients have a size to read. \(W\) is constant in θ and \(U\) goes as \(\tan^2\theta\), so a number of degrees taken off either is a number at one chosen angle. \(U = 1.0\ \deg^2\) alone is 0.268° FWHM at \(2\theta = 30\)° and 3.172° at \(2\theta = 145\)°, a factor of twelve across an ordinary pattern.
A bound declared in degrees is hard to transfer for the same reason. \(X\) carries \(\max = 1.0\)°, which is 79.4 Å on Cu Kα against 21.3 Å on 11-BM’s 0.4139 Å, so one cap admits a 3.7× spread in the physics. The sample terms \(X_s\) and \(P\) carry \(\min = 0\) and no maximum. Decide such a limit as a size, and report it in degrees.
6.3. Reading a width as a strain¶
Microstrain broadens by a fixed fraction of every d-spacing rather than by a fixed reciprocal-space width [Stokes and Wilson, 1944]. Differentiating Bragg’s law \(\lambda = 2d\sin\theta\) at fixed \(\lambda\) gives \(\Delta d/d = -\cot\theta\, \Delta\theta\), so a relative spread in \(d\) appears as
Source: rietx.model.profiles.caglioti
which is the \(\tan\theta\) term of (3.3). Inverting it for the coefficient \(Y_s\) in deg \(2\theta\),
Source: rietx.model.profiles.caglioti.microstrain_from_strain_coefficient
reading \(\sqrt{U_s}\) in place of \(Y_s\) for the Gaussian variance coefficient of (3.1), exactly as (6.3) reads \(\sqrt{P}\).
No wavelength appears in (6.5), and no shape constant
either. The contrast with (6.3) is the whole of the
asymmetry. Six sample-broadening quantities appear in (3.1) and
(3.3): two sizes, two strains, and the Stephens block of
(6.8). The two named “size” are the ones that depend on \(\lambda\).
One specimen measured at two wavelengths therefore shows the same number of
degrees of strain broadening, and size coefficients in the ratio
\(\lambda_2 / \lambda_1\), with their Gaussian variances in the ratio squared. A
joint refinement shares the size rather than the coefficient.
rietx.params.multi normalises it and reports the rescaling as the
SIZE_NORMALISED_ACROSS_WAVELENGTHS diagnostic.
\(\Delta d/d\) here is the FWHM of the d-spacing distribution, matching what
(6.4) relates and what the coefficients of
(3.1)–(3.3) are. Other codes publish other
measures of the same width. GSAS-II’s mustrain is \(2\Delta d/d\) in units of
\(10^{-6}\), and FullProf’s apparent strain is \(\tfrac{1}{2}\beta^{*}d\), read off
the integral breadth rather than the FWHM. A microstrain is comparable between
codes only with its convention attached.
6.4. Stephens anisotropic strain¶
Stephens’ phenomenological model [Stephens, 1999] lets every crystallite carry its own lattice metric. With
the spread of a quadratic form’s coefficients makes the variance of \(M\) a homogeneous quartic in \((h, k, l)\):
Source: rietx.crystallography.stephens.sigma2_m
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
Source: rietx.crystallography.stephens.strain_width_deg
added to the Lorentzian FWHM as \(\Lambda(hkl)\cdot\tan\theta\). It is the first width in the model to depend on \(hkl\). Stephens writes this quantity \(\Gamma_A(hkl)\), with the \(\tan\theta\) inside it. \(\Lambda\) is this manual’s letter for the coefficient alone, because \(\Gamma\) is already the FWHM in Peak profiles. FullProf calls the same thing \(D_{ST}\).
Warning
Three independent labelling conventions sit behind these \(S_{HKL}\), and getting any one wrong rescales every published number. Transfer a literature value without them and the width law is wrong while the fit still refines.
\(\sqrt{\sum S\cdot\text{monomial}}\cdot d^2\cdot 10^{-6}\) is the FWHM of the \(\Delta M/M = 2\Delta d/d\) distribution, and not its standard deviation. No \(\sqrt{8\ln 2}\) appears anywhere. That is Stephens’ own convention rather than a later simplification: the note under his equation (4) says the \(\sqrt{8\ln 2}\) between the r.m.s. and the FWHM has been folded into the definition of \(S_{HKL}\). A published set that did not fold it in differs by that one constant on every coefficient.
The coefficients are carried in 10⁻¹² Å⁻⁴ rather than physical Å⁻⁴. That choice is numerical: 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.
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 \(3(h^2k^2 + h^2l^2 + k^2l^2)\), so their printed values differ by small integer factors as well. Check a literature \(S_{HKL}\) numerically before transferring it.
6.5. 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. It is the rank-4 twin of the rank-2 construction used for ADPs [Peterse and Palm, 1966], and shares 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, and a set
outside the allowed subspace raises rather than being symmetrised. A Stephens
block locks the scalar lor_strain, whose isotropic direction is identically
that column, so the block subsumes it. Free the block in the sample-broadening
stage itself.
6.6. The positivity cone, the seed and the guard¶
\(\sigma^2(M) \ge 0\) for every \(hkl\) is a cone coupling all fifteen coefficients,
so 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 with nothing left to report
(Estimation). Read a firing as “these coefficients are not quotable”,
never as evidence of anisotropy. The guard’s test is one-sided, because zero
lies on the cone itself. 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 it is both the seed and the only legal start. At \(S \equiv 0\) the square root in (6.8) has unbounded slope, so strain stages seed through a dedicated mechanism rather than a generic parameter seed.