Heating expands the sample support, lifting the surface off the goniometer axis by micrometres — and every peak shifts by Δ2θ = −2s cosθ/R. Zoom in on a low-angle and a high-angle reflection and watch which one suffers.
Displacement s0 µm
Radius R
Reading the panels: grey dashed peaks sit at the true Bragg angles; blue peaks are what the detector records with the sample lifted by s. The cosθ factor makes the low-angle reflection move the most — the exact region where phase identification lives. A larger goniometer radius R dilutes the same s into a smaller angular error.
s: 0 µmT: 25 °CΔ2θ @ 25°: 0.000°Δ2θ @ 65°: 0.000°
Schematic teaching model. Peak shifts follow the first-order specimen-displacement relation Δ2θ = −(2s cosθ/R)·(180/π), with s and R in the same units; peak shapes are idealized Gaussians and the heat-ramp mapping of temperature to support growth (25→600 °C giving +150 µm) is illustrative of a representative stage, not a universal constant. Real instruments add further aberrations — sample transparency, axial divergence, flat-specimen error — not shown here. The internal-standard correction is drawn as ideal, removing the displacement term exactly; real corrections are limited by the standard's own accuracy. Playback is time-compressed.