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Sandbox Physics

E17 · Thin-foil orientation mapping

TKD: Through the Nanocrystal

Compare reflection with an off-axis transmission screen, then balance foil thickness, counts and binning to recover nanograin orientation and phase.

Interactive modelTKD: Through the Nanocrystal
Selected stored pattern counts—\text{—}
Selected detector pixel counts—\text{—}
Indexed one-axis rotation—\text{—}
Indexed teaching phase—\text{—}
Best candidate correlation—\text{—}
Best versus runner-up score gap—\text{—}
Accepted fraction of all pixels—\text{—}
Separate phase and angle input audit—\text{—}
Accepted-pixel mean angular error—\text{—}
Assumed footprint FWHM along tilt—\text{—}
Incident dose per scan pixel—\text{—}
Experiment target—\text{—}

Physics tutorial

TKD: Through the Nanocrystal

BackgroundCompare reflection with an off-axis transmission screen, then balance foil thickness, counts and binning to recover nanograin orientation and phase.

Why it mattersTarget: index at least 90 percent of pixels, recover the correct phase and rotation within 2 degrees in at least 90 percent of all pixels, and keep accepted-pixel mean angular error within 2 degrees. In TKD, also keep the assumed footprint FWHM at most 12 nm.

Start with the essentials

Focus question
Can a thinner specimen resolve smaller grains?
One-sentence intuition
An indexing result depends on calibrated geometry, adequate counts and a suitable candidate family.

Core mathematical model

Cone edges on the detector

dhkl=a/h2+k2+l2∣s^⋅n^hkl∣=sin⁡θB=λ/(2dhkl)\begin{aligned}d_{hkl}&=a/\sqrt{h^2+k^2+l^2}\\|\widehat{\mathbf s}\cdot\widehat{\mathbf n}_{hkl}|&=\sin\theta_B=\lambda/(2d_{hkl})\end{aligned}

Unit rays intersect the two Bragg cones. Their central plane traces form the band center lines; band intensity is prescribed.

Count first, index second

μj=Iτe η Bj∑kBkc^=arg max⁡c  (N−N‾)⋅(Tc−T‾c)∥N−N‾∥ ∥Tc−T‾c∥\begin{aligned}\mu_j&=\frac{I\tau}{e}\,\eta\,\frac{B_j}{\sum_k B_k}\\\widehat c&=\underset{c}{\operatorname{arg\,max}}\;\frac{(\mathbf N-\overline N)\cdot(\mathbf T_c-\overline T_c)}{\|\mathbf N-\overline N\|\,\|\mathbf T_c-\overline T_c\|}\end{aligned}

Each scan point stores a detector pattern. Search the declared orientation and phase dictionary using normalized correlation of those counts.

Declared foil response

p=t(1 nm)cos⁡ασ=(1 nm)4+(0.065p)2f=1−e−p/15η=0.08fe−p/150\begin{aligned}p&=\frac{t}{(1\,{\rm nm})\cos\alpha}\\\sigma&=(1\,{\rm nm})\sqrt{4+(0.065p)^2}\\f&=1-e^{-p/15}\\\eta&=0.08f e^{-p/150}\end{aligned}

Normal thickness becomes a beam-path thickness. A narrower footprint can still deliver too few counts; these coefficients are teaching assumptions.

Common difficulties

Map color is not a full orientation

Typical misconceptionA beam IPF map alone proves every orientation and phase.

Better mental modelThe search is restricted to the declared one-axis family and fictional phases. Treat dark pixels and small candidate margins as limitations, and inspect the stored patterns.

Run the experiment

  1. 01

    Predict before indexing

    Select a stored pixel and inspect its detector pattern. Compare the independently labeled input.

    What to observe: Band positions encode orientation; band widths also depend on the lattice spacing.
  2. 02

    Calibrate and search

    Try the wrong-center preset, then set the center error to zero. Sweep binning without reacquiring.

    What to observe: Bad geometry can reject a clean pattern; coarse bins erase width differences.
  3. 03

    Read the maps

    Compare beam IPF, phases, connected grains and neighbor disorientation. Select a boundary pixel.

    What to observe: Unindexed pixels stay dark; grain segmentation uses recovered data.
  4. 04

    Test the tradeoff

    Compare the same nanograin specimen in reflection and transmission. Sweep foil thickness and exposure.

    What to observe: Thin foils sharpen the assumed footprint; low counts and thick foils can still defeat indexing.