Incondensed matter physics, theLyddane–Sachs–Teller relation (orLST relation) determines the ratio of the natural frequency of longitudinal optic lattice vibrations (phonons) (${\displaystyle {\omega }_{\text{LO}}}$) of anionic crystal to the natural frequency of the transverse optical lattice vibration (${\displaystyle {\omega }_{\text{TO}}}$) for long wavelengths (zero wavevector).^{[1][2][3][4][5]} The ratio is that of the static permittivity${\displaystyle {\epsilon }_{\text{st}}}$ to the permittivity for frequencies in the visible range${\displaystyle {\epsilon }_{\mathrm{\infty }}}$.^[6]

The relation holds for systems with a single optical branch, such as cubic systems with two different atoms per unit cell. For systems with many phonon branches, the relation does not necessarily hold, as the permittivity for any pair of longitudinal and transverse modes will be altered by the other modes in the system. The Lyddane–Sachs–Teller relation is named after the physicists R. H. Lyddane,Robert G. Sachs, andEdward Teller.

The Lyddane–Sachs–Teller relation applies to optical lattice vibrations that have an associated netpolarization density, so that they can produce long rangedelectromagnetic fields (over ranges much longer than the inter-atom distances). The relation assumes an idealized polar ("infrared active") optical lattice vibration that gives a contribution to the frequency-dependentpermittivity described by a lossless Lorentzian oscillator:

${\displaystyle \epsilon (\omega )=\epsilon (\mathrm{\infty })+(\epsilon (\mathrm{\infty })-{\epsilon }_{st})\frac{{\omega }_{\text{TO}}^2}{{\omega }^2-{\omega }_{\text{TO}}^2},}$

where${\displaystyle \epsilon (\mathrm{\infty })}$ is the permittivity at high frequencies,${\displaystyle {\epsilon }_{st}}$ is the static DC permittivity, and${\displaystyle {\omega }_{\text{TO}}}$ is the "natural" oscillation frequency of the lattice vibration taking into account only the short-ranged (microscopic) restoring forces.

The above equation can be plugged intoMaxwell's equations to find the complete set of normal modes including all restoring forces (short-ranged and long-ranged), which are sometimes calledphonon polaritons. These modes are plotted in the figure. At everywavevector there are three distinct modes:

• alongitudinal wave mode occurs with an essentially flat dispersion at frequency${\displaystyle {\omega }_{\text{LO}}}$.

• In this mode, the electric field is parallel to the wavevector and produces no transverse currents, hence it is purely electric (there is no associated magnetic field).
• The longitudinal wave is basically dispersionless, and appears as a flat line in the plot at frequency${\displaystyle {\omega }_{\text{LO}}}$. This remains 'split off' from the bare oscillation frequency even at high wave vectors, because the importance of electric restoring forces does not diminish at high wavevectors.

• twotransverse wave modes appear (actually, four modes, in pairs with identical dispersion), with complex dispersion behavior.

• In these modes, the electric field is perpendicular to the wavevector, producing transverse currents, which in turn generate magnetic fields. As light is also a transverse electromagnetic wave, the behaviour is described as a coupling of the transverse vibration modes with thelight inside the material (in the figure, shown as red dashed lines).
• At high wavevectors, the lower mode is primarily vibrational. This mode approaches the 'bare' frequency${\displaystyle {\omega }_{\text{TO}}}$ because magnetic restoring forces can be neglected: the transverse currents produce a small magnetic field and the magnetically induced electric field is also very small.
• At zero, or low wavevector theupper mode is primarily vibrational and its frequency instead coincides with the longitudinal mode, with frequency${\displaystyle {\omega }_{\text{LO}}}$. This coincidence is required by symmetry considerations and occurs due toelectrodynamic retardation effects that make the transverse magnetic back-action behave identically to the longitudinal electric back-action.^[7]

The longitudinal mode appears at the frequency where thepermittivity passes through zero, i.e.${\displaystyle \epsilon ({\omega }_{\text{LO}})=0}$. Solving this for the Lorentzian resonance described above gives the Lyddane–Sachs–Teller relation.^[3]

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 (also known as damping or dissipation).
• Materials may have multiple phonon resonances that add together to produce the permittivity.
• There may be other electrically active degrees of freedom (notably, mobile electrons) and non-Lorentzian oscillators.

In the case of multiple, lossy Lorentzian oscillators, there are generalized Lyddane–Sachs–Teller relations available.^[8]Most generally, the permittivity cannot be described as a combination of Lorentizan oscillators, and the longitudinal mode frequency can only be found as acomplex zero in the permittivity function.^[8]

The most general Lyddane–Sachs–Teller relation applicable in crystals where the phonons are affected byanharmonic damping has been derived in Ref.^[9] and reads as

the absolute value is necessary since the phonon frequencies are now complex, with an imaginary part that is equal to the finite lifetime of the phonon, and proportional to the anharmonic phonon damping (described by Klemens' theory for optical phonons).

A corollary of the LST relation is that for non-polar crystals, the LO and TO phonon modes aredegenerate, and thus${\displaystyle {\epsilon }_{\text{st}}={\epsilon }_{\mathrm{\infty }}}$. This indeed holds for the purely covalent crystals of thegroup IV elements, such as for diamond (C), silicon, and germanium.^[10]

In the frequencies between${\displaystyle {\omega }_{\text{TO}}}$ and${\displaystyle {\omega }_{\text{LO}}}$ there is 100% reflectivity. This range of frequencies (band) is called theReststrahl band. The name derives from the Germanreststrahl which means "residual ray".^[11]

The static and high-frequency dielectric constants ofNaCl are${\displaystyle {\epsilon }_{\text{st}}=5.9}$ and${\displaystyle {\epsilon }_{\mathrm{\infty }}=2.25}$, and the TO phonon frequency${\displaystyle {\nu }_{\text{TO}}}$ is${\displaystyle 4.9}$ THz. Using the LST relation, we are able to calculate that^[12]

${\displaystyle {\nu }_{\text{LO}}=\sqrt{{\epsilon }_{\text{st}}/{\epsilon }_{\mathrm{\infty }}}\times {\nu }_{\text{TO}}=7.9}$ THz

One of the ways to experimentally determine${\displaystyle {\omega }_{\text{TO}}}$ and${\displaystyle {\omega }_{\text{LO}}}$ is throughRaman spectroscopy.^[13][14] 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 (${\displaystyle k=0}$) of theBrillouin zone. This is also the point where the photon-phonon coupling most often occurs for theStokes shift measured in Raman. Hence two peaks will be present in theRaman spectrum, each corresponding to the TO and LO phonon frequency.

• Reststrahlen effect

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• Ashcroft, Neil W.; Mermin, N. David (1976).Solid State Physics (1 ed.).Holt, Rinehart and Winston.ISBN 978-0030839931.
• Klingshirn, Claus F. (2012).Semiconductor Optics (4 ed.). Springer.ISBN 978-364228362-8.
• Fox, Mark (2010).Optical Properties of Solids (2 ed.).Oxford University Press.ISBN 978-0199573370.
• Robinson, L. C. (1973).Physical Principles of Far-Infrared Radiation, Volume 10 (1 ed.). Elsevier.ISBN 978-0080859880.

• Irmer, G.; Wenzel, M.; Monecke, J. (1996). "The temperature dependence of the LO(T) and TO(T) phonons in GaAs and InP".Physica Status Solidi B.195 (1):85–95.Bibcode:1996PSSBR.195...85I.doi:10.1002/pssb.2221950110.ISSN 0370-1972.
• Lyddane, R.; Sachs, R.; Teller, E. (1941). "On the Polar Vibrations of Alkali Halides".Physical Review.59 (8):673–676.Bibcode:1941PhRv...59..673L.doi:10.1103/PhysRev.59.673.
• Chang, I. F.; Mitra, S. S.; Plendl, J. N.; Mansur, L. C. (1968). "Long-Wavelength Longitudinal Phonons of Multi-Mode Crystals".Physica Status Solidi B.28 (2):663–673.Bibcode:1968PSSBR..28..663C.doi:10.1002/pssb.19680280224.
• Casella, L.; Zaccone, A. (2021). "Soft mode theory of ferroelectric phase transitions in the low-temperature phase".Journal of Physics: Condensed Matter.33 (16): 165401.arXiv:2103.10262.doi:10.1088/1361-648X/abdb68.PMID 33440364.

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