Phonon Fit Window Three-Row Scan - Condensed Matter and Plasma Physics Step by Step

A phonon-fit-window scan varies the frequency window across three rows, checking that the harmonic-oscillator phonon dispersion stays consistent inside the window row by row. The arithmetic is exact only for the stated fit windows, frequencies, and rows in the ideal harmonic lattice model. Real measurements add defects, strain, temperature, finite crystal size, instrument calibration, peak fitting choices, finite precision, and measurement uncertainty.

Wavelength moves below, onto, and above a standing-mode fit. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.

highlighted = computed this step

Short wavelength misses the standing window

Mode index stays fixed while wavelength moves through the exact fit window. The fit bit is checked by the rendered mode source.

\begin{array}{c}n\lambda=3\cdot 6=18\ \mathrm{m}\ \ne\ 24\ \mathrm{m}\\fit=0,\quad nodes=4\end{array}

Matching wavelength fits the standing window

Mode index stays fixed while wavelength moves through the exact fit window. The fit bit is checked by the rendered mode source.

\begin{array}{c}n\lambda=3\cdot 8=24\ \mathrm{m}\ =\ 24\ \mathrm{m}\\fit=1,\quad nodes=4\end{array}

Long wavelength overshoots the standing window

Mode index stays fixed while wavelength moves through the exact fit window. The fit bit is checked by the rendered mode source.

\begin{array}{c}n\lambda=3\cdot 10=30\ \mathrm{m}\ \ne\ 24\ \mathrm{m}\\fit=0,\quad nodes=4\end{array}