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An ideal cubic perovskite structure is obtained by putting La at the cube corners, Re at the body center, and N at the face centers. In fractional crystal coordinates one convenient basis isThe counting is $8/8=1$ La, one Re and $6/2=3$ N per cell. Corner-sharing $\mathrm{Re}{\mathrm{N}}_6$ octahedra form the network, with La in the larger interstices. The ideal cell illustrates the connectivity; octahedral tilting can lower the actual crystal symmetry.

There are two chemical atoms in the primitive unit cell, hence six displacement coordinates and six phonon branches at each wavevector. Three rigid translations cost no restoring potential energy in the long-wavelength limit. ConsequentlyFor propagation along $[100]$, one acoustic phonon has longitudinal polarization along $x$ and two have transverse polarization along $y,z$. The cubic crystal symmetry makes those two transverse directions equivalent. In terms of the elastic stiffness tensor and the mass density,Thus there are two distinct acoustic velocities in general, one longitudinal and a doubly degenerate transverse value. This degeneracy of normal modes concerns the two transverse polarizations; it does not imply equality with the longitudinal speed of sound.

In the diamond cubic structure of silicon, the longitudinal acoustic phonon branch and longitudinal optical phonon branch meet at $X=(2\pi /a,0,0)$. The two atomic basis sites are identical, and the associated crystal symmetry protects their boundary degeneracy of normal modes. In aluminium phosphide, the two sites contain different chemical atoms. The corresponding zinc blende structure lacks the site-exchange symmetry, and these two longitudinal branches generally split at $X$.

A simple way to see the splitting is a diatomic-chain dispersion relation with equal springs, representing successive $[100]$ atomic planes by alternating masses $M_1,M_2$ at spacing $d=a/4$. With effective spring constant $K$, its angular frequencies obeyAt $X$, $kd=\pi /2$, so the inter-sublattice coupling cancels and the two squared angular frequencies are $2K/M_1$ and $2K/M_2$. Equal masses give a contact; unequal masses giveIn aluminium phosphide, the heavier phosphorus plane moves in the lower-frequency edge mode, and the lighter aluminium plane in the higher-frequency mode in this model. Their similar atomic masses explain why the splitting is relatively small. Different electronic environments can also change the force constant matrix. The chain calculation illustrates the mechanism; a quantitative three-dimensional phonon spectrum requires its full dynamical matrix. It does not assert that all transverse acoustic and optical branches meet in silicon.

Barium titanate has a high-temperature cubic perovskite structure, with Ti centred in its oxygen octahedron on average. This phase is paraelectric. Near room temperature the structure is tetragonal, with relative ionic displacements that give a spontaneous electric polarization. This phase is ferroelectric: an applied electric field can switch between differently polarized domains. Both phases have the same composition, BaTiO${}_3$; the transition near $120{}^{\circ }\mathrm{C}$ changes crystal symmetry and electrical response.