Folded wave-number counting for a monatomic chain
= Folded wave-number counting for a monatomic chain
{title2=$g_{|q|}=Na/\pi$}
A periodic chain of $N$ sites and length $Na$ has allowed signed <wave numbers> separated by $2\pi/(Na)$. The continuum state density on the signed axis is $Na/(2\pi)$. Combining opposite propagation directions onto $0<|q|<\pi/a$ doubles it to $Na/\pi$. This counts the same $N$ modes, not two independent <acoustic phonon branches>.