Phonon Mode Energy Three-Row Scan - Solid-State Physics: Lattices, Phonons, and Band Filling Step by Step

Standing-mode rows connect wavelength, frequency, and packet energy. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.

highlighted = computed this step

Second mode has the longest wavelength row

Mode index 2 on a 12 m chain gives wavelength 12 m, frequency four-thirds Hz, and four-packet energy 16 J.

n\lambda=2\cdot 12=24\ \mathrm{m},\quad f=4/3\ \mathrm{Hz},\quad E=16\ \mathrm{J}

Third mode is the middle wavelength row

Mode index 3 on a 12 m chain gives wavelength 8 m, frequency 2 Hz, and four-packet energy 24 J.

n\lambda=3\cdot 8=24\ \mathrm{m},\quad f=2\ \mathrm{Hz},\quad E=24\ \mathrm{J}

Fourth mode raises frequency and energy

Mode index 4 on a 12 m chain gives wavelength 6 m, frequency eight-thirds Hz, and four-packet energy 32 J.

n\lambda=4\cdot 6=24\ \mathrm{m},\quad f=8/3\ \mathrm{Hz},\quad E=32\ \mathrm{J}