Lyddane–Sachs–Teller Relation
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In
condensed matter physics Condensed matter physics is the field of physics that deals with the macroscopic and microscopic physical properties of matter, especially the solid and liquid State of matter, phases, that arise from electromagnetic forces between atoms and elec ...
, the Lyddane–Sachs–Teller relation (or LST relation) determines the ratio of the natural frequency of longitudinal optic lattice vibrations (
phonon A phonon is a collective excitation in a periodic, elastic arrangement of atoms or molecules in condensed matter, specifically in solids and some liquids. In the context of optically trapped objects, the quantized vibration mode can be defined a ...
s) (\omega_\text) of an ionic crystal to the natural frequency of the transverse optical lattice vibration (\omega_\text) for long wavelengths (zero wavevector). The ratio is that of the static permittivity \varepsilon_ to the permittivity for frequencies in the visible range \varepsilon_. 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 Robert G. Sachs (May 4, 1916 – April 14, 1999) was an American theoretical physicist, a founder and a director of the Argonne National Laboratory. Sachs was also notable for his work in theoretical nuclear physics, terminal ballistics, and nu ...
, and
Edward Teller Edward Teller (; January 15, 1908 – September 9, 2003) was a Hungarian and American Theoretical physics, theoretical physicist and chemical engineer who is known colloquially as "the father of the hydrogen bomb" and one of the creators of ...
.


Origin and limitations

The Lyddane–Sachs–Teller relation applies to optical lattice vibrations that have an associated net
polarization density In classical electromagnetism, polarization density (or electric polarization, or simply polarization) is the vector field that expresses the volumetric density of permanent or induced electric dipole moments in a dielectric material. When a die ...
, so that they can produce long ranged
electromagnetic fields In physics, electromagnetism is an interaction that occurs between particles with electric charge via electromagnetic fields. The electromagnetic force is one of the four fundamental forces of nature. It is the dominant force in the interacti ...
(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-dependent
permittivity In electromagnetism, the absolute permittivity, often simply called permittivity and denoted by the Greek letter (epsilon), is a measure of the electric polarizability of a dielectric material. A material with high permittivity polarizes more ...
described by a lossless Lorentzian oscillator: : \varepsilon(\omega) = \varepsilon(\infty) + (\varepsilon(\infty)-\varepsilon_)\frac, where \varepsilon(\infty) is the permittivity at high frequencies, \varepsilon_ is the static DC permittivity, and \omega_\text 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 into
Maxwell's equations Maxwell's equations, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, Electrical network, electr ...
to find the complete set of normal modes including all restoring forces (short-ranged and long-ranged), which are sometimes called phonon polaritons. These modes are plotted in the figure. At every
wavevector In physics, a wave vector (or wavevector) is a vector used in describing a wave, with a typical unit being cycle per metre. It has a magnitude and direction. Its magnitude is the wavenumber of the wave (inversely proportional to the wavelength), ...
there are three distinct modes: * a
longitudinal wave Longitudinal waves are waves which oscillate in the direction which is parallel to the direction in which the wave travels and displacement of the medium is in the same (or opposite) direction of the wave propagation. Mechanical longitudinal ...
mode occurs with an essentially flat dispersion at frequency \omega_\text. :* 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 \omega_\text. 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. * two
transverse wave In physics, a transverse wave is a wave that oscillates perpendicularly to the direction of the wave's advance. In contrast, a longitudinal wave travels in the direction of its oscillations. All waves move energy from place to place without t ...
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 the
light Light, visible light, or visible radiation is electromagnetic radiation that can be visual perception, perceived by the human eye. Visible light spans the visible spectrum and is usually defined as having wavelengths in the range of 400– ...
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 \omega_\text 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 the ''upper'' mode is primarily vibrational and its frequency instead coincides with the longitudinal mode, with frequency \omega_\text. This coincidence is required by symmetry considerations and occurs due to electrodynamic retardation effects that make the transverse magnetic back-action behave identically to the longitudinal electric back-action. The longitudinal mode appears at the frequency where the
permittivity In electromagnetism, the absolute permittivity, often simply called permittivity and denoted by the Greek letter (epsilon), is a measure of the electric polarizability of a dielectric material. A material with high permittivity polarizes more ...
passes through zero, i.e. \varepsilon(\omega_\text) = 0. 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 (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. 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.


Anharmonic crystals

The most general Lyddane–Sachs–Teller relation applicable in crystals where the phonons are affected by anharmonic damping has been derived in Ref. 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).


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 \varepsilon_\text=\varepsilon_\infty. This indeed holds for the purely covalent crystals of the group IV elements, such as for diamond (C), silicon, and germanium.


Reststrahlen effect

In the frequencies between \omega_\text and \omega_\text there is 100% reflectivity. This range of frequencies (band) is called the Reststrahl band. The name derives from the German ''reststrahl'' which means "residual ray".


Example with NaCl

The static and high-frequency dielectric constants of
NaCl Sodium chloride , commonly known as edible salt, is an ionic compound with the chemical formula NaCl, representing a 1:1 ratio of sodium and chloride ions. It is transparent or translucent, brittle, hygroscopic, and occurs as the mineral hali ...
are \varepsilon_\text=5.9 and \varepsilon_\infty=2.25, and the TO phonon frequency \nu_\text is 4.9 THz. Using the LST relation, we are able to calculate that :::\nu_\text=\sqrt\times\nu_\text=7.9 THz


Experimental methods


Raman spectroscopy

One of the ways to experimentally determine \omega_\text and \omega_\text is through
Raman spectroscopy Raman spectroscopy () (named after physicist C. V. Raman) is a Spectroscopy, spectroscopic technique typically used to determine vibrational modes of molecules, although rotational and other low-frequency modes of systems may also be observed. Ra ...
. 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 (k=0) of the
Brillouin zone In mathematics and solid state physics, the first Brillouin zone (named after Léon Brillouin) is a uniquely defined primitive cell in reciprocal space Reciprocal lattice is a concept associated with solids with translational symmetry whic ...
. This is also the point where the photon-phonon coupling most often occurs for the Stokes shift measured in Raman. Hence two peaks will be present in the Raman spectrum, each corresponding to the TO and LO phonon frequency.


See also

* Reststrahlen effect


Citations


References


Textbooks

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Articles

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