18.1 Electromagnetic radiation and matter interaction

Properties of Electromagnetic Radiation

Fundamental Characteristics

Electromagnetic (EM) radiation consists of oscillating electric and magnetic fields that propagate through space at the speed of light. These two fields are perpendicular to each other and to the direction of propagation, making EM radiation a transverse wave.

Three quantities characterize any EM wave, and they're all connected:

• Wavelength ($\lambda$): the distance between successive wave crests
• Frequency ($\nu$): the number of oscillations per second (units of Hz or $\text{s}^{-1}$)
• Energy ($E$): the energy carried per photon

The core relationship linking them is:

$E = h\nu = \frac{hc}{\lambda}$

where $h$ is Planck's constant ($6.626 \times 10^{-34}\ \text{J·s}$) and $c$ is the speed of light in vacuum ($2.998 \times 10^{8}\ \text{m/s}$). Notice that energy scales with frequency but inversely with wavelength. High-frequency radiation carries more energy per photon.

Interaction with Matter

When EM radiation encounters matter, three outcomes are possible:

• Absorption: The photon's energy matches the energy gap between two allowed quantum states of the atom or molecule. The photon is destroyed and the system is promoted to a higher energy state.
• Emission: An excited atom or molecule drops from a higher energy state to a lower one, releasing a photon whose energy equals the gap between those states.
• Scattering: Photons are redirected by matter without a net electronic transition. Rayleigh scattering preserves the photon's frequency, while Raman scattering shifts it (more on this below).

All three processes are governed by quantum mechanics. Energy levels in atoms and molecules are discrete, not continuous, so only photons with specific energies can be absorbed or emitted. Which transitions are actually allowed is determined by selection rules that enforce conservation of energy, angular momentum, and symmetry constraints on the wavefunctions involved.

Absorption and Emission of Radiation

Absorption Process

A molecule absorbs a photon when the photon energy exactly matches the spacing between two allowed energy levels:

$\Delta E = E_{\text{upper}} - E_{\text{lower}} = h\nu$

This promotes the system from the lower state to the upper (excited) state. The excited state is less stable and has higher potential energy than the ground state. Typical excited-state lifetimes range from nanoseconds ($\sim 10^{-9}\ \text{s}$) for electronic states to microseconds or longer, depending on the type of transition and the molecular environment.

Emission Process

The reverse of absorption: an excited system releases a photon as it relaxes to a lower energy state. Emission comes in two forms:

1. Spontaneous emission: The excited state decays on its own, emitting a photon with energy $h\nu = \Delta E$.
2. Stimulated emission: An incoming photon with energy $h\nu = \Delta E$ triggers the transition. The emitted photon has the same frequency, phase, polarization, and direction as the stimulating photon. This coherence is the operating principle behind lasers.

Spectra and the Boltzmann Distribution

Absorption and emission give rise to characteristic spectra that serve as fingerprints for chemical identification:

• Emission spectra show discrete bright lines or bands at wavelengths corresponding to allowed downward transitions.
• Absorption spectra show dark lines or bands where photons have been removed from a continuous source by the sample.

The intensity of spectral lines depends on how many molecules occupy the initial state. The Boltzmann distribution quantifies this:

$\frac{N_j}{N_i} = \frac{g_j}{g_i} \exp\!\left(-\frac{\Delta E}{k_BT}\right)$

where $N_j$ and $N_i$ are the populations of the upper and lower states, $g_j/g_i$ is the ratio of their degeneracies, $k_B$ is Boltzmann's constant, and $T$ is the absolute temperature. At higher temperatures, more molecules populate excited states, which increases emission intensity and can alter relative absorption line strengths.

Frequency, Wavelength, and Energy

Relationship between Frequency and Wavelength

Frequency and wavelength are inversely related through the speed of light:

$c = \lambda\nu$

Higher frequency means shorter wavelength, and vice versa. The electromagnetic spectrum, ordered from low to high energy:

Energy of Electromagnetic Radiation

Because $E = h\nu = hc/\lambda$, the energy per photon increases as you move from radio waves toward gamma rays. This has direct physical consequences:

• High-energy photons (X-rays, gamma rays) can ionize atoms by ejecting core or valence electrons.
• Low-energy photons (radio waves, microwaves) lack the energy to ionize and instead drive rotational or spin transitions.

Spectroscopists often express photon energy in wavenumbers ($\tilde{\nu} = 1/\lambda$, in units of $\text{cm}^{-1}$), which is directly proportional to energy and convenient for IR and Raman work.

Transitions in Matter-Radiation Interactions

Electronic Transitions

Electronic transitions involve moving an electron between different energy levels (orbitals) in an atom or molecule.

• Excitation: An electron absorbs a photon and jumps from a lower-energy orbital to a higher-energy orbital.
• De-excitation: The electron drops back down, emitting a photon.

These transitions typically fall in the UV-visible region because the energy gaps between electronic states are relatively large (on the order of 1–10 eV). Common examples include excitation of valence electrons in atoms (the basis of atomic emission spectroscopy) and $\pi \to \pi^*$ or $n \to \pi^*$ transitions in organic chromophores.

Vibrational Transitions

Vibrational transitions involve changes in the vibrational energy of a molecule and are driven by infrared radiation. Molecules vibrate in distinct modes: bond stretching (symmetric and asymmetric) and bending (scissoring, rocking, wagging, twisting).

The vibrational frequency depends on two factors:

• The masses of the bonded atoms (lighter atoms vibrate faster)
• The force constant (bond strength) of the bond (stronger bonds vibrate faster)

This is captured by the harmonic oscillator model: $\nu = \frac{1}{2\pi}\sqrt{k/\mu}$, where $k$ is the force constant and $\mu$ is the reduced mass.

IR spectroscopy exploits these transitions. Different functional groups absorb at characteristic frequencies (e.g., O-H stretch near $3200\text{–}3600\ \text{cm}^{-1}$, C=O stretch near $1700\ \text{cm}^{-1}$). A vibration is IR-active only if it produces a change in the molecule's dipole moment during the motion.

Rotational Transitions

Rotational transitions involve changes in a molecule's rotational energy and are probed by microwave radiation. The energy gaps between rotational levels are much smaller than vibrational or electronic gaps.

Rotational energy levels for a rigid rotor are:

$E_J = BJ(J+1), \quad B = \frac{\hbar^2}{2I}$

where $J$ is the rotational quantum number, $B$ is the rotational constant, and $I$ is the moment of inertia. Because $I$ depends on atomic masses and bond lengths, microwave spectroscopy provides precise measurements of molecular geometry. Isotopic substitution changes the mass and therefore $I$, shifting the spectrum and allowing isotope-specific structural analysis.

A molecule must possess a permanent dipole moment to be microwave-active (the selection rule $\Delta J = \pm 1$ applies for linear molecules).

Scattering and Luminescence

Raman scattering occurs when an incident photon interacts with a molecule and exchanges energy with a vibrational mode, producing a scattered photon at a shifted frequency. Unlike IR absorption, Raman activity requires a change in the molecule's polarizability during the vibration. This makes Raman and IR spectroscopy complementary: vibrations that are IR-inactive (e.g., symmetric stretches in homonuclear diatomics like $\text{N}_2$) can be Raman-active, and vice versa.

Fluorescence and phosphorescence are both forms of photoluminescence:

• Fluorescence: The molecule absorbs a UV/visible photon, reaches a singlet excited state ($S_1$), undergoes rapid vibrational relaxation, then emits a lower-energy photon as it returns to $S_0$. This happens fast (nanosecond timescale) because the transition is spin-allowed ($\Delta S = 0$).
• Phosphorescence: After reaching $S_1$, the molecule undergoes intersystem crossing to a triplet state ($T_1$). Emission from $T_1 \to S_0$ is spin-forbidden, so it's much slower (milliseconds to seconds).

Both processes emit photons at longer wavelengths (lower energy) than the absorbed photon. The energy difference between absorbed and emitted photons is called the Stokes shift. Applications include fluorescence microscopy, chemical sensing, and organic light-emitting diodes (OLEDs).