en.wikipedia.org/wiki/Phonon

The article says these gaps are "too small for radiation to occur," but modern nanoscale heat-transfer literature says the opposite: radiative heat transfer is very important at nanometre and sub-nanometre separations.

A 2015 Nature Communications paper specifically studied the transition from near-field thermal radiation to phonon heat conduction at sub-nanometre gaps. Its abstract states that for gaps larger than 1 nm, the conductance agrees with continuum fluctuational electrodynamics (a radiative model), and that as the gap shrinks below 1 nm, low-frequency acoustic phonons add extra channels while the distinction between radiation and conduction becomes blurred. In other words, these distances are not simply "too small for radiation to occur"; rather, radiation remains part of the physics and there is a crossover/overlap regime.

A review in Nanoscale Horizons likewise explains that when separations are comparable to or smaller than the dominant thermal wavelength, standard far-field radiation ideas fail, but near-field radiative heat transfer still occurs and can be dominated by diffraction, interference, and electromagnetic-wave tunneling. So the problem is not that radiation cannot occur, but that classical far-field radiation models are insufficient.

So the precise correction is: at nanometre-scale gaps, heat transfer is not forbidden to occur by radiation; instead, near-field radiative transfer can occur, and at very small gaps it may coexist with or give way to phonon-mediated transfer.

The phrase "N ≥ 2 different atoms" is too restrictive and is false as stated. What matters for optical branches is that the primitive cell contains more than one atom (or basis degree of freedom), not that the atoms be of different chemical species.

A standard counterexample is diamond. Diamond’s crystal structure has two atoms in the basis, and those two atoms are both carbon. Yet diamond has well-known long-wavelength optical phonons measured by Raman scattering.

So the correct condition is essentially: in a 3D crystal with N atoms in the primitive cell, there are 3 acoustic branches and 3N−3 optical branches. The atoms do not have to be different elements.

This is especially important because readers could wrongly infer that elemental crystals such as diamond or silicon cannot have optical phonons, which is not true.

This sentence overgeneralizes a property that is true only for long-wavelength acoustic modes (near (k \to 0), i.e. near the Brillouin-zone center).

Open instructional sources distinguish the long-wavelength limit from finite-wavelength acoustic modes:

• LibreTexts states that for the long-wavelength limit, the displacements of the atoms in the acoustic branch have the same amplitude, direction, and phase.
• A UCSC simulation page explicitly contrasts a very long wavelength acoustic phonon, where adjacent-atom spacing is unchanged, with a short-wavelength acoustic phonon, where there is a significant phase difference because of the finite wavelength.

So the blanket statement "In acoustic modes, all the p atoms vibrate in phase" is not correct for the whole acoustic branch. A more accurate version would be that acoustic modes correspond to in-phase motion in the long-wavelength limit; at finite wavevector, relative phases and displacements generally need not remain exactly the same.