Mixed Miller Index Spacing Scan - Biophysical Instrumentation II: From Signals to Genomes Step by Step

A mixed Miller index scan checks that a fixed twelve-meter schematic lattice constant still divides exactly when all three Miller indices are nonzero at once, not just a single scaled axis: index triple (2,3,6) gives spacing 12/7 meters, (1,4,8) gives 4/3 meters, and (3,4,12) gives 12/13 meters, each exact because the sum of squared indices is itself a perfect square. Every earlier Miller-index row in this book held two of the three indices at zero; real crystal planes are genuinely three-dimensional, and this scan shows the same exact-division rule holding for three real index triples, not just the single-axis case.

Every earlier Miller-index row in this book held two indices at zero; three genuinely mixed index triples still divide the same lattice constant exactly. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.

highlighted = computed this step

Chapter six's rows all kept two indices at zero

Every earlier Miller-index row in this book held k and l at zero and only changed h. A real plane family can carry all three indices at once: h=2, k=3, l=6 still gives an exact spacing because 49 is a perfect square.

h^{2}+k^{2}+l^{2}=49=7^{2},\quad d=\tfrac{12}{7}\ \text{m}

Three genuinely three-dimensional plane families

Each row uses a different exact Pythagorean-style index triple, not just a scaled single axis. The spacing still divides the same lattice constant exactly in every row.

\begin{array}{c|c|c|c}h&k&l&d\\2&3&6&\tfrac{12}{7}\ \text{m}\\1&4&8&\tfrac{4}{3}\ \text{m}\\3&4&12&\tfrac{12}{13}\ \text{m}\\\end{array}

A larger index triple still divides the lattice exactly

The third row uses h=3, k=4, l=12. Even with larger indices, 169 is still an exact perfect square, so the spacing stays exact.

d=\frac{12\ \text{m}}{\sqrt{169}}=\tfrac{12}{13}\ \text{m}