2. Peak positions

2.1. Lattice metric and Bragg’s law

The d-spacing of reflection \((h,k,l)\) follows from the reciprocal metric tensor \(G^*\) [Shm01]:

(2.1)\[\frac{1}{d^2} \;=\; \mathbf{h} \cdot G^* \cdot \mathbf{h}^\top, \qquad G^* = G^{-1},\]

Source: rietx.crystallography.lattice

where \(G\) is the direct metric tensor built from \((a, b, c, \alpha, \beta, \gamma)\). Peak positions then follow Bragg’s law,

(2.2)\[2\theta \;=\; 2 \arcsin\!\left(\frac{\lambda}{2d}\right).\]

Source: rietx.crystallography.lattice

Every emission line diffracts at its own Bragg angle. Differentiating (2.2) at fixed \(d\) gives the doublet-splitting law

(2.3)\[\Delta 2\theta \;=\; 2 \tan\theta \cdot \frac{\Delta\lambda}{\lambda},\]

Source: rietx.schemas.instrument

which grows with \(\tan\theta\) — a Kα₂ line is never a fixed offset from Kα₁.

2.2. Aberration shifts

Additive \(2\theta\) shifts with distinct angular signatures are modelled; the signatures are what makes them separable, and only barely so (the decorrelation workflow below). Which shifts exist depends on the geometry, because each is derived for one specimen shape: the two below the zero-point error are flat-plate aberrations, and the capillary has its own pair further down.

The zero-point error is a constant, and is the only one common to every geometry. Sample displacement in Bragg-Brentano geometry, for a flat specimen whose surface sits a distance \(s\) off the goniometer axis with goniometer radius \(R\) [KA74, Wil63]:

(2.4)\[\Delta 2\theta \;=\; -\frac{2 s}{R} \cos\theta \quad [\mathrm{rad}].\]

Source: rietx.model.corrections.displacement_shift_deg

The \(\cos\theta\) dependence is what separates it from the zero-point error. Sample transparency — finite beam penetration puts the effective diffracting surface below the physical one (thick-sample limit [KA74, Wil63]):

(2.5)\[\Delta 2\theta \;=\; -t \sin 2\theta \quad [\mathrm{rad}], \qquad t = \frac{1}{2 \mu_{\mathrm{eff}} R},\]

Source: rietx.model.corrections.transparency_shift_deg

with \(t \ge 0\) dimensionless; for strongly absorbing samples \(t \to 0\) and the correction vanishes.

These three columns (constant, \(\cos\theta\), \(\sin 2\theta\)) are nearly collinear over a typical angular range, and all three trade against the cell parameters. The house workflow decorrelates them by calibration: refine zero and displacement on a standard whose certified cell is held fixed, save the instrument profile, and load it frozen for sample work.

Capillary displacement is the Debye-Scherrer counterpart, for a capillary whose diffracting volume sits off the centre of the \(2\theta\) circle [MVDC+99]:

(2.6)\[\Delta 2\theta \;=\; \frac{-a \sin 2\theta + b \cos 2\theta}{R} \quad [\mathrm{rad}].\]

Source: rietx.model.corrections.capillary_displacement_shift_deg

Here \(a\) is the displacement along the incident beam, positive downstream, and \(b\) the displacement perpendicular to it in the diffraction plane, positive toward increasing \(2\theta\). The paper prints the same expression as \((x \sin 2\theta - y \cos 2\theta)/R\) and draws no axes; the signs above are fixed by derivation, and other codes attach the letter \(x\) to the other term, so the shapes are what carries the meaning. Both are exactly zero when the capillary is centred, and both are held fixed unless the geometry declares \(R\).

The trio for this geometry is therefore (constant, \(\sin 2\theta\), \(\cos 2\theta\)), and it is separable for the same reason and to the same limited degree: over \(5\)–\(160°\) the smallest eigenvalue of the unit-column Gram matrix is \(5.2 \times 10^{-2}\), and over \(5\)–\(25°\) it is \(1.1 \times 10^{-5}\).

2.3. Wavelength scales

Kα₁/Kα₂ wavelengths are peak positions of the measured line shapes, not centroids, quoted on one consistent scale: the NIST X-ray Transition Energies Database [DKI+03, NIST05], whose 3d-metal values derive from the Hölzer et al. measurements [HolzerFD+97] and whose Mo/Ag values from Deslattes & Kessler [DK85] — one column is the claim, not one paper. One column of one evaluation for all anodes is the load-bearing choice — mixing wavelength scales between anodes (or against an older table) is the classic ~100 ppm cell-parameter error. Bearden’s compilation [Bea67] is a different scale (Mo Kα₂ differs by 24 ppm); individual rows must not be “corrected” toward it.

Source: rietx.schemas.instrument