# Physics of Light: Absorption#

From MIT Lecture Notes: A Classical Model for Spectroscopy

Why does light absorb? The quantum mechanical model is a bit opaque and doesn’t tell us much. Let’s use the classical model instead.

Molecules are composed of charged particles. Light (an E&M field) exerts a force on these charges. Force exerted on the molecules depends on the field strength, magnitude of the charges, and how far the charges move.

## Classical Model: Uses three things#

• Light: an oscillating electric magnetic field
• Matter: we treat it as a harmonic oscilattor
• Interactions: oscillating external force field driving a harmonic oscillator

### 1. Light#

$\overline{\mathrm{E}} (\overline{\mathrm{r}} ,\mathrm{t} )=\hat{\varepsilon } (\overline{\mathrm{r}} )\mathrm{E}_{\mathrm{o}}\cos (\omega \mathrm{t} -\overline{\mathrm{k}} \cdot \overline{\mathrm{r}} -\phi )$
$\hat{\varepsilon } (\overline{\mathrm{r}} )\ \Longrightarrow \ polarization\ vector$
$\mathrm{E}_{\mathrm{o}} \Longrightarrow amplitude$
$\omega \mathrm{t} \Longrightarrow frequency\ ( rad/s)$
$\overline{\mathrm{k}} \Longrightarrow wavevector\ defines\ direction\ of\ propogation$

Let’s simplify:

1. Let’s propogate along $$\vec{x}$$ and drop polarization so $$\phi=0$$
$\mathrm{E}(\mathrm{x}, \mathrm{t})=\mathrm{E}_{0} \cos (\omega \mathrm{t}-\mathrm{kx})$

We can also drop the wave vector $$(|\mathrm{k}| \rightarrow 0, \text { since } \lambda>>\mathrm{x} \text { and we consider molecules at } \mathrm{x}=0)$$

$\mathrm{E}(\mathrm{t})=\mathrm{E}_{0} \cos \omega \mathrm{t}$
$I=\frac{c}{4 \pi}\left|E_{0}\right|^{2}$
$\begin{array}{l}{|\mathrm{k}|=\frac{2 \pi}{\lambda}=\frac{\omega}{\mathrm{c}}} \\ {\mathrm{c}=2.998 \times 10^{8} \mathrm{m} / \mathrm{s}}\end{array}$

### 2. Molecules: treat as harmonic oscillator#

Why should we be able to call molecules harmonic oscillators? i.e., a mass on a spring?

Molecules feel a restoring force when pushed from equilibrium.

The covalent bond can be thought of as a spring. The equilibrium length is a balance between attractive and repulsive forces. \ If we push/pull on this bond, there is a restoring force that pushes the system back to equilibrium.

This analogy works for other systems also

• Electronic states—think of pushing electron clouds away from equilibrium distribution. (for instance, benzene pi orbitals)
• Magnetic resonance—In a magnetic field, magnetic spin moments—nuclear spins—align with field.\ If we push a spin away from field, it will want to relax back.

Summary:

• Scattering from Damped Harmonic Oscillator
• Assume that a molecule is a simple harmonic oscillator with a single harmonic oscillation frequency $$\omega_{0}(2 \pi v)$$
• When irradiated by monochromatic electromagnetic wave of frequencyω0,theelectromagneticwaveoffrequencyω0,theelectron undergoes an acceleration, while the nucleus, being massive, is assumed not to move