Solid-State Boundary and Invariant Scans

Lattice Fixed-Length Three-Row Scan

A lattice fixed-length scan varies the number of repeats across three rows at fixed lattice spacing, checking that L = N * a holds row by row. The arithmetic is exact only for the stated lattice spacings, repeat counts, and rows in the ideal infinite-crystal model. Real measurements add defects, strain, temperature, finite crystal size, instrument calibration, peak fitting choices, finite precision, and measurement uncertainty.

Cell count and spacing trade off while total length stays fixed. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.

highlighted = computed this step

Coarse cells keep the same total length

The cell count and spacing trade off while total length stays 12 m. The reciprocal row is 1/6 per m.

L=2⋅6=12 m,b=1/6 per m,sites=3L=2\cdot 6=12\ \mathrm{m},\quad b=1/6\ \mathrm{per\ m},\quad sites=3
Fixed-length lattice row coarseThe reciprocal spacing cites the same lattice source.a=6 mb=1/6 1/ma*b=1

Middle cells keep the same total length

The cell count and spacing trade off while total length stays 12 m. The reciprocal row is 1/4 per m.

L=3⋅4=12 m,b=1/4 per m,sites=4L=3\cdot 4=12\ \mathrm{m},\quad b=1/4\ \mathrm{per\ m},\quad sites=4
Fixed-length lattice row middleThe reciprocal spacing cites the same lattice source.a=4 mb=1/4 1/ma*b=1

Fine cells keep the same total length

The cell count and spacing trade off while total length stays 12 m. The reciprocal row is 1/2 per m.

L=6⋅2=12 m,b=1/2 per m,sites=7L=6\cdot 2=12\ \mathrm{m},\quad b=1/2\ \mathrm{per\ m},\quad sites=7
Fixed-length lattice row fineThe reciprocal spacing cites the same lattice source.a=2 mb=1/2 1/ma*b=1