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Diffstat (limited to 'source/know/concept/lyddane-sachs-teller-relation')
| -rw-r--r-- | source/know/concept/lyddane-sachs-teller-relation/index.md | 45 |
1 files changed, 24 insertions, 21 deletions
diff --git a/source/know/concept/lyddane-sachs-teller-relation/index.md b/source/know/concept/lyddane-sachs-teller-relation/index.md index 9cec9dc..60c8984 100644 --- a/source/know/concept/lyddane-sachs-teller-relation/index.md +++ b/source/know/concept/lyddane-sachs-teller-relation/index.md @@ -20,7 +20,7 @@ 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: +and the phonon has $$\Omega$$ and $$K$$, then: $$\begin{aligned} \hbar \omega @@ -71,8 +71,8 @@ $$\begin{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, +The latter depends on the displacement between the ions, +instead of their displacement from their equilibrium position, so we need to write $$\vb{x}_{+} - \vb{x}_{-}$$ instead of $$\vb{x}_{+}$$. Respectively dividing the equations by $$m_{+}$$ and $$m_{-}$$ @@ -95,10 +95,10 @@ $$\begin{aligned} \end{aligned}$$ 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 $$K \approx 0$$. +This is close enough, because IR light waves are much larger +than the crystal's unit cell, so we can ignore all spatial variation in $$\vb{E}$$ +(the [electric dipole approximation](/know/concept/electric-dipole-approximation/)), +which is equivalent to assuming that $$K = 0$$. For the sake of generality, we also introduce an empirical damping rate $$\gamma$$, @@ -147,7 +147,9 @@ $$\begin{aligned} \end{aligned}$$ In the limits of low and high frequencies $$\omega$$, -we see that $$\varepsilon_r$$ is higher in the former: +we see that $$\varepsilon_r$$ is higher in the former +(also recall that $$\chi_\mathrm{low} > \chi_\mathrm{high}$$ +according to the original Lorentz oscillator model): $$\begin{aligned} \varepsilon_{\mathrm{low}} @@ -159,7 +161,8 @@ $$\begin{aligned} = 1 + \chi_\mathrm{high} \end{aligned}$$ -We can use these quantities to rewrite the relative permittivity $$\varepsilon_r$$ as follows: +We can use these quantities to rewrite +the relative permittivity $$\varepsilon_r$$ as follows: $$\begin{aligned} \varepsilon_r(\omega) @@ -167,9 +170,8 @@ $$\begin{aligned} \frac{\Omega_\mathrm{TO}^2}{\Omega_\mathrm{TO}^2 - \omega^2 - i \gamma \omega} \end{aligned}$$ -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: +For weak damping $$\gamma \approx 0$$, there exists a frequency +with zero permittivity, which we call $$\Omega_\mathrm{LO}$$: $$\begin{aligned} 0 @@ -179,8 +181,8 @@ $$\begin{aligned} \end{aligned}$$ The physical significance of $$\varepsilon_r = 0$$ can be -seen from [Gauss' law](/know/concept/maxwells-equations), under the assumption that there is -no net charge density: +seen from [Gauss' law](/know/concept/maxwells-equations), +under the assumption that there is no net charge density: $$\begin{aligned} \nabla \cdot \vb{D} @@ -188,10 +190,10 @@ $$\begin{aligned} = 0 \end{aligned}$$ -If $$\varepsilon_r \neq 0$$, then $$\nabla \cdot \vec{E} = 0$$, +If $$\varepsilon_r \neq 0$$, then $$\nabla \cdot \vb{E} = 0$$, 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. +However, if $$\varepsilon_r = 0$$, then $$\nabla \cdot \vb{E} \neq 0$$, +representing a longitudinal electric wave, analogous to plasmons in metals. Rearranging the equation for $$\Omega_\mathrm{LO}$$ gives us the **Lyddane-Sachs-Teller (LST) relation**: @@ -230,17 +232,18 @@ 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: +with a self-referential dispersion relation given by: $$\begin{aligned} \omega_\mathrm{pp}(K) - = \frac{c}{\sqrt{\varepsilon_r(\omega_\mathrm{pp})}} K + = \frac{c}{\sqrt{\varepsilon_r(\omega_\mathrm{pp}(K))}} K \end{aligned}$$ 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. +But now, plotting $$\omega_\mathrm{pp}(K)$$ reveals that there is no intersection! +This is a good example of the typical *anti-crossing* +behavior of strongly coupled quantum systems. |