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+---
+title: "Lyddane-Sachs-Teller relation"
+sort_title: "Lyddane-Sachs-Teller relation"
+date: 2024-04-15
+categories:
+- Physics
+layout: "concept"
+---
+
+While the [Lorentz oscillator model](/know/concept/lorentz-oscillator-model/)
+originally studied the electric dipole formed by an electron and its nucleus,
+it can also be applied to the nuclei of polar crystals,
+i.e. crystals held together by polar bonds between ions.
+When an [electromagnetic wave](/know/concept/electromagnetic-wave-equation/)
+passes by, its [electric field](/know/concept/electric-field/)
+$$\vb{E}(t)$$ exerts a force on the ions, leading to an optical response.
+
+We are talking about light waves (photons)
+creating lattice vibrations (phonons),
+i.e. a photon-phonon conversion,
+where the total energy and momentum must be conserved.
+If the photon has frequency $$\omega$$ and wavenumber $$k$$,
+and the phonon $$\Omega$$ and $$K$$, then:
+
+$$\begin{aligned}
+    \hbar \omega
+    = \hbar \Omega
+    \qquad \qquad
+    \hbar k
+    = \hbar K
+\end{aligned}$$
+
+In other words, such a conversion can only take place
+at intersections of the dispersion relations $$\omega(k)$$ and $$\Omega(K)$$.
+The latter consists of two branches:
+low-frequency *acoustic* modes and higher-frequency *optical* modes.
+Meanwhile, the photon dispersion is simply $$\omega = c k / n$$,
+where $$n$$ is the medium's refractive index.
+
+For acoustic phonons, the dispersions only intersect at $$k = K = 0$$,
+which is simply a static solid in a static electric field.
+For optical phonons, the intersection is at a nonzero $$k$$.
+In addition, light is a transverse wave,
+so it can only interact with transverse phonons,
+meaning that we must only consider **transverse optical (TO) phonons**.
+
+A wave's group velocity is the slope of its dispersion,
+so $$\ipdv{\omega}{k}$$ and $$\ipdv{\Omega}{K}$$ in this case.
+Clearly, light is much faster than sound,
+so $$\omega(k)$$ is much steeper than $$\Omega(K)$$,
+meaning that the photon-phonon conversion
+will happen at relatively low $$k$$.
+In practice, the intersection is in the infrared (IR),
+hence TO phonons are sometimes called **IR active**.
+
+We consider a 1D chain of unit cells along the $$z$$-axis,
+each containing a positive and a negative ion
+oscillating transversely along the $$x$$-axis.
+For optical phonon modes, the ions always move in opposite directions.
+Let the ions have masses $$m_{-}$$ and $$m_{+}$$,
+then the Lorentz oscillator model tells us
+that the displacements $$\vb{x}_{+}(t)$$ and $$\vb{x}_{-}(t)$$ are governed by:
+
+$$\begin{aligned}
+    m_{+} \dvn{2}{\vb{x}_{+}}{t}
+    &= - \kappa (\vb{x}_{+} - \vb{x}_{-}) + q \vb{E}
+    \\
+    m_{-} \dvn{2}{\vb{x}_{-}}{t}
+    &= - \kappa (\vb{x}_{-} - \vb{x}_{+}) - q \vb{E}
+\end{aligned}$$
+
+Where $$\vb{E}(t) = \vb{E}_0 e^{- i \omega t}$$ represents the light,
+and $$\kappa$$ is the spring constant of the polar bonds' restoring force.
+Note that the latter depends on the displacement between the ions,
+instead of from their equilibrium position,
+so we need to write $$\vb{x}_{+} - \vb{x}_{-}$$ instead of $$\vb{x}_{+}$$.
+
+Respectively dividing the equations by $$m_{+}$$ and $$m_{-}$$
+and subtracting the latter from the former,
+we arrive at the following combined equation,
+where $$m$$ is the [reduced mass](/know/concept/reduced-mass/):
+
+$$\begin{aligned}
+    \dvn{2}{}{t} (\vb{x}_{+} - \vb{x}_{-})
+    = - \frac{\kappa}{m} (\vb{x}_{+} - \vb{x}_{-}) + \frac{q}{m} \vb{E}
+\end{aligned}$$
+
+Defining the relative displacement $$\vb{x} \equiv \vb{x}_{+} \!-\! \vb{x}_{-}$$,
+and recognizing that $$\kappa / m$$ is the TO phonons'
+natural resonance frequency $$\Omega_\mathrm{TO}^2$$:
+
+$$\begin{aligned}
+    \dvn{2}{\vb{x}}{t} + \Omega_\mathrm{TO}^2 \vb{x}
+    = \frac{q}{m} \vb{E}
+\end{aligned}$$
+
+Note that $$\Omega_\mathrm{TO}$$ is the phonon frequency for $$K = 0$$.
+This is because IR light waves are much larger than the crystal's unit cell,
+so we are ignoring all spatial variation in $$\vb{E}$$
+(i.e. the [electric dipole approximation](/know/concept/electric-dipole-approximation/)).
+This is equivalent to assuming that $$K \approx 0$$.
+
+For the sake of generality,
+we also introduce an empirical damping rate $$\gamma$$,
+like in the original Lorentz oscillator model:
+
+$$\begin{aligned}
+    \dvn{2}{\vb{x}}{t} + \gamma \dv{\vb{x}}{t} + \Omega_\mathrm{TO}^2 \vb{x}
+    = \frac{q}{m} \vb{E}
+\end{aligned}$$
+
+Inserting the ansatz $$\vb{x}(t) = \vb{x}_0 e^{- i \omega t}$$
+and isolating for the amplitude $$\vb{x}_0$$, we find:
+
+$$\begin{aligned}
+    \vb{x}_0
+    = \frac{q \vb{E}_0}{m (\Omega_\mathrm{TO}^2 - \omega^2 - i \gamma \omega)}
+\end{aligned}$$
+
+The induced polarization density $$\vb{P}$$ is then the sum
+of the electrons' and ions' contributions $$\vb{P}_e$$ and $$\vb{P}_i$$.
+The former is described by a background susceptibility $$\chi$$,
+and the latter by each unit cell's dipole moment $$\vb{p} = q \vb{x}$$
+multiplied by the number of cells per unit volume $$N$$:
+
+$$\begin{aligned}
+    \vb{P}
+    \approx \varepsilon_0 \chi \vb{E} + N q \vb{x}
+    = \bigg( \varepsilon_0 \chi + \frac{N q^2}{m (\Omega_\mathrm{TO}^2 - \omega^2 - i \gamma \omega)} \bigg) \vb{E}
+\end{aligned}$$
+
+Note that we are neglecting how each dipole shields its neighbors.
+This approximation can be improved afterwards by using
+the [Clausius-Mossotti relation](/know/concept/clausius-mossotti-relation/).
+
+With our expression for $$\vb{P}$$, we can find
+the [dielectric function](/know/concept/dielectric-function/) $$\varepsilon_r(\omega)$$
+using the definition of the electric displacement field
+$$\vb{D} = \varepsilon_0 \vb{E} + \vb{P} = \varepsilon_0 \varepsilon_r \vb{E}$$,
+yielding:
+
+$$\begin{aligned}
+    \boxed{
+        \varepsilon_r(\omega)
+        = 1 + \chi(\omega) + \frac{N q^2}{\varepsilon_0 m (\Omega_\mathrm{TO}^2 - \omega^2 - i \gamma \omega)}
+    }
+\end{aligned}$$
+
+In the limits of low and high frequencies $$\omega$$,
+we see that $$\varepsilon_r$$ is higher in the former:
+
+$$\begin{aligned}
+    \varepsilon_{\mathrm{low}}
+    &= \, \lim_{\omega \to 0} \, \varepsilon_r(\omega)
+    = 1 + \chi_\mathrm{low} + \frac{N q^2}{\varepsilon_0 m \Omega_\mathrm{TO}^2}
+    \\
+    \varepsilon_{\mathrm{high}}
+    &= \lim_{\omega \to \infty} \varepsilon_r(\omega)
+    = 1 + \chi_\mathrm{high}
+\end{aligned}$$
+
+We can use these quantities to rewrite the relative permittivity $$\varepsilon_r$$ as follows:
+
+$$\begin{aligned}
+    \varepsilon_r(\omega)
+    = \varepsilon_{\mathrm{high}} + (\varepsilon_{\mathrm{low}} - \varepsilon_{\mathrm{high}})
+    \frac{\Omega_\mathrm{TO}^2}{\Omega_\mathrm{TO}^2 - \omega^2 - i \gamma \omega}
+\end{aligned}$$
+
+For weak damping $$\gamma \approx 0$$, there exists a frequency,
+which we will call $$\Omega_\mathrm{LO}$$ in anticipation,
+where the dielectric function is zero:
+
+$$\begin{aligned}
+    0
+    = \varepsilon_r(\Omega_\mathrm{LO})
+    = \varepsilon_{\mathrm{high}}
+    + (\varepsilon_{\mathrm{low}} - \varepsilon_{\mathrm{high}}) \frac{\Omega_\mathrm{TO}^2}{\Omega_\mathrm{TO}^2 - \Omega_\mathrm{LO}^2}
+\end{aligned}$$
+
+The physical significance of $$\varepsilon_r = 0$$ can be
+seen from [Gauss' law](/know/concept/maxwells-equations), under the assumption that there is
+no net charge density:
+
+$$\begin{aligned}
+    \nabla \cdot \vb{D}
+    = \varepsilon_0 \varepsilon_r \nabla \cdot \vb{E}
+    = 0
+\end{aligned}$$
+
+If $$\varepsilon_r \neq 0$$, then $$\nabla \cdot \vec{E} = 0$$,
+corresponding to a transverse light wave as usual.
+However, if $$\varepsilon_r = 0$$, then $$\nabla \cdot \vec{E} \neq 0$$,
+representing a longitudinal electric wave, like a plasmon in metal.
+Rearranging the equation for $$\Omega_\mathrm{LO}$$
+gives us the **Lyddane-Sachs-Teller (LST) relation**:
+
+$$\begin{aligned}
+    \boxed{
+        \frac{\Omega_\mathrm{LO}^2}{\Omega_\mathrm{TO}^2}
+        = \frac{\varepsilon_{\mathrm{low}}}{\varepsilon_{\mathrm{high}}}
+    }
+\end{aligned}$$
+
+$$\Omega_\mathrm{LO}$$ is the natural frequency
+of such **longitudinal optical (LO) phonons** for $$K = 0$$.
+Recall that only transverse phonons interact with light:
+the significance of this result is that we can measure
+$$\varepsilon_\mathrm{low}$$, $$\varepsilon_\mathrm{high}$$,
+and $$\Omega_\mathrm{TO}$$ with light,
+and use that to calculate a quantity for an effect
+that we cannot interact with directly.
+The caveat is that this is only valid for simple polar crystals.
+
+For $$\omega$$-values between $$\Omega_\mathrm{TO}$$ and $$\Omega_\mathrm{LO}$$,
+the permittivity $$\varepsilon_r$$ is negative,
+meaning the reflectivity $$R$$ equals $$1$$,
+i.e. the material becomes a perfect reflector:
+
+$$\begin{aligned}
+    R
+    = \bigg| \frac{i \sqrt{|\varepsilon_r|} - 1}{i \sqrt{|\varepsilon_r|} + 1} \bigg|^2
+    = \frac{|\varepsilon_r|^2 + 1^2}{|\varepsilon_r|^2 + 1^2}
+    = 1
+\end{aligned}$$
+
+This region of 100% reflectivity is called the **Reststrahlen band**.
+In practice, real materials have $$\gamma > 0$$, which reduces $$R$$ somewhat.
+
+Because the photons and TO phonons interact so strongly
+for $$\omega \approx \Omega_\mathrm{TO}$$,
+they can be treated as a single **phonon polariton** there,
+with a dispersion relation given by:
+
+$$\begin{aligned}
+    \omega_\mathrm{pp}(K)
+    = \frac{c}{\sqrt{\varepsilon_r(\omega_\mathrm{pp})}} K
+\end{aligned}$$
+
+Earlier, when treating the photon and phonon separately,
+we wanted the intersection between $$\omega(k)$$ and $$\Omega(K)$$.
+But now, for $$\omega_\mathrm{pp}(K)$$, there is none! This is a good example
+of the typical *anti-crossing* behavior of strongly coupled systems.
+
+
+
+## References
+1.  M. Fox,
+    *Optical properties of solids*, 2nd edition,
+    Oxford.