In condensed matter physics, the Lyddane–Sachs–Teller relation determines the ratio of the natural frequency of longitudinal optic lattice vibrations of an ionic crystal to the natural frequency of the transverse optical lattice vibration for long wavelengths. The ratio is that of the staticpermittivity to the permittivity for frequencies in the visible range. The Lyddane–Sachs–Teller relation is named after the physicists R. H. Lyddane, Robert G. Sachs, and Edward Teller. . The separation between LO and TO phonon frequencies near the Γ-point is described by the LST relation. Note this plot shows much higher wavevectors than considered below, and the scale cannot not show the hybridization of the TO branch with light.
Origin and limitations
The Lyddane–Sachs–Teller relation applies to optical lattice vibrations that have an associated net polarization density, so that they can produce long ranged electromagnetic fields. The relation assumes an idealized polar optical lattice vibration that gives a contribution to the frequency-dependent permittivity described by a lossless Lorentzian oscillator: where is the permittivity at high frequencies, is the static polarizability of the optical lattice mode, and is the "natural" oscillation frequency of the lattice vibration taking into account only the short-ranged restoring forces. of phonon polaritons in GaP. Red curves are the uncoupled phonon and photon dispersion relations, black curves are the result of coupling. The LST relation relates the frequencies of the horizontal red curve and the black curve intercept at k=0. The above equation can be plugged into Maxwell's equations to find the complete set of normal modes including all restoring forces, which are sometimes called phonon polaritons. These modes are plotted in the figure. At every wavevector there are three distinct modes:
two transverse wave modes appear, with complex dispersion behavior.
The longitudinal mode appears at the frequency where the permittivity passes through zero, i.e.. Solving this for the Lorentzian resonance described above gives the Lyddane–Sachs–Teller relation. Since the Lyddane–Sachs–Teller relation is derived from the lossless Lorentzian oscillator, it may break down in realistic materials where the permittivity function is more complicated for various reasons:
Real phonons have losses.
Materials may have multiple phonon resonances that add together to produce the permittivity.
There may be other electrically active degrees of freedom and non-Lorentzian oscillators.
In the case of multiple, lossy Lorentzian oscillators, there are generalized Lyddane–Sachs–Teller relations available. Most generally, the permittivity cannot be described as a combination of Lorentizan oscillators, and the longitudinal mode frequency can only be found as a complex zero in the permittivity function.
Non-polar crystals
A corollary of the LST relation is that for non-polar crystals, the LO and TO phonon modes are degenerate, and thus. This indeed holds for the purely covalent crystals of the group IV elements, such as for diamond, silicon, and germanium.
In the frequencies between and there is 100% reflectivity. This range of frequencies is called the Reststrahl band. The name derives from the German reststrahl which means "residual ray".
The static and high-frequency dielectric constants of NaCl are and, and the TO phonon frequency is THz. Using the LST relation, we are able to calculate that
One of the ways to experimentally determine and is through Raman spectroscopy. As previously mentioned, the phonon frequencies used in the LST relation are those corresponding to the TO and LO branches evaluated at the gamma-point of the Brillouin zone. This is also the point where the photon-phonon coupling most often occurs for the Stokes shiftmeasured in Raman. Hence two peaks will be present in the Raman spectrum, each corresponding to the TO and LO phonon frequency.
