Condon model

In optics and materials science, Condon model is a mathematical formula for the frequency dependence of the chirality parameter of bi-isotropic or bi-anisotropic media. It was reported by Edward Condon, William Altar and Henry Eyring in 1937 in its definitive form,^[1][2] with its earlier forms being introduced by Max Born, Heinrich Gerhard Kuhn and Léon Rosenfeld, among others.^[3]

Edward Condon c. late 1940s

Mathematical formulation

Electric and magnetic constitutive relations for a dispersive and reciprocal chiral material are written as:^[4]

${\displaystyle {\textbf {D}}=\varepsilon (\omega ){\textbf {E}}-i\kappa (\omega ){\sqrt {\varepsilon (\omega )\mu (\omega )}}{\textbf {H}}}$

${\displaystyle {\textbf {B}}=\mu (\omega ){\textbf {H}}+i\kappa (\omega ){\sqrt {\varepsilon (\omega )\mu (\omega )}}{\textbf {E}}}$

where ${\textstyle \varepsilon (\omega )}$ and ${\textstyle \mu (\omega )}$ are the frequency-dependent permittivity and magnetic susceptibility. ${\textstyle \kappa (\omega )}$ denotes the chirality parameter for magnetoelectric coupling. Using a quantum mechanical treatment of molecular transitions that facilitate chiral behavior, Condon et al. arrives at a single oscillator oscillator expression for the chirality parameter, known as "the one‐electron rotatory power":^[1][2][4]

${\displaystyle \kappa (\omega )={\frac {\omega R}{\omega ^{2}-\omega ^{2}-i\omega \Gamma }}}$

where

• ${\textstyle \omega _{0}}$ is the angular resonant frequency of the molecular transition.
• ${\textstyle \Gamma }$ is the damping term.
• ${\textstyle R}$ is the rotational strength of the molecular transition.

Alternatively, an expression with multiple oscillators can be used to denote multiple molecular transition between the states${\textstyle a}$ to ${\textstyle b}$:^[5]

${\displaystyle \kappa (\omega )=\sum _{b}{\frac {\omega R_{ba}}{\omega _{0,ba}^{2}-\omega ^{2}-i\omega \Gamma _{ba}}}}$

Under passivity constraints, imaginary parts of the complex Condon expression and the other constitutive paremeters obey the inequality:^[4]

${\displaystyle {\text{Im}}\left[\kappa (\omega )\right]^{2}<{\text{Im}}\left[\varepsilon (\omega )\right]\;{\text{Im}}\left[\mu (\omega )\right]c_{0}^{2}}$

where ${\textstyle c_{0}}$ is the speed of light in vacuum. The model is often approximated with a single-pole oscillator whose resonance lies far away from other molecular transitions. The presence of angular frequency (${\textstyle \omega }$) term in the numerator suggests the absence of chirality in the static limit.^[4] Since the model is causal and thus obeys the Kramers–Kronig relations,^[6] it is used in the time-domain analytical and numerical modeling of wave propagation in chiral media.^{[7][8][9][10]}

Condon model parameters of chiral materials such as glucose solutions and metamaterials can be retrieved from experimental measurements of optical rotatory dispersion^[6] and electromagnetic simulation data.^[11]

See also

• Franck–Condon principle
• Lorentz oscillator model