Electromagnetic Radiation

Introduction to Electromagnetic Radiation

We base our description of the properties of radiation on both a wave and a particle view of the properties. According to the classical view electromagnetic radiation travels through free space as a wave that we can express as E = E₀e^-iwt where E₀ is an amplitude and e^-iwt is an oscillatory function.

Blackbody radiation

At the end of nineteenth century the wave view of electromagnetic radiation was confronted with an experimentally observed frequency spectrum for the radiation emitted from a black body. A black body is an idealization used to describe the temperature-dependent emission of radiation. When we speak of metal becoming "red hot" as it is heated we are referring to the fact that it emits visible radiation. As the metal is heated further, it turns "white hot". Thus, as the metal is heated it emits light of higher frequency. The frequency distribution can be measured and has the appearance shown below.

The energy used to heat the metal is emitted as radiation. In the real world there is also loss of energy by conduction and convection but we ignore that here. The point here is that the wave theory (classical physics) fails to correctly describe this effect. The approach taken was to consider a black body as a cavity at some temperature, T. If the cavity has length, L then there will be allowed waves provided 2ml = L. This is just the condition for a standing wave to exist in the cavity. However, this simple idea predicts that there will be more modes for shorter wavelengths. In other words the density of cavity modes increases without bound! This disagrees with experiment and is known as the ultraviolet catastrophe.

Planck realized that a single assumption concerning the radiation in the cavity could produce agreement the experimental result. We define the wave vector k = 2p/l. In three dimensions the boundary conditions require

where n, m, and l are integers. The mode density N_k is

The factor of two arises from the fact that there are two polarizations possible for each mode. The density of frequencies is N_n

The mode density is defined as the number of modes per unit volume per unit frequency interval. Dividing by the volume V = L³ and taking the derivative with respect to n, we obtain

So far we are following the classical derivation that leads to the ultraviolet catastrophe. Planck hypothesized that the modes of electromagnetic radiation could only exist in multiples of hn, where h is a constant. The energy levels in a blackbody are thus E_n = nhn. The probability that the nth level is occupied is given by the Boltzmann probability distribution

The average energy is given by

Taking this average energy per mode and multiplying it by the mode density we obtain the Planck distribution law for blackbody radiation.

This distribution law agrees with experiment. The distribution has the form shown below.

The photoelectric effect

It is most convenient to consider absorption of light by viewing it as a particle with a discrete (quantum) of energy. In this description the light behave as a particle with energy E = hn where h is Planck’s constant and n is the frequency. This is described as the photoelectric effect. The photoelectric effect is an ionization process and therefore requires that incident radiation have a minimum (threshold) energy. If the energy of the incident photon hn > E_ionization then an electron is ejected (and the metal is ionized). Such an ionization process can occur in a molecule as well along with a host of absorptive transitions that will be the subject of this course. The photoelectric effect is the original experiment that gave rise to the particle view of electromagnetic radiation. The effect can be understood only if we view an incident photon as a particle that can have energy sufficient to eject an electron.

The view that light is both a wave and a particle can be accomplished by picturing light as a wave packet. The packet has finite extent and contains a large number of vibrations within its spatial envelope.

The electromagnetic spectrum

The wavelength of visible light is of the order 400 – 780 nm whereas molecular dimensions are of the order of 1 nm. Clearly, the interaction of the "wave" with the molecule is non-classical and the absorption of light occurs because of a particle-like process in which a photon is absorbed by the relatively small molecule. It is important to keep this in mind since in much of the following we discuss the wave picture. The wave picture is necessary in order to understand the polarization of light and its properties when it interacts with an interface.

In discussing molecular spectroscopy we are interested in light in the microwave region (rotational motion), infrared region (vibrational motion), and visible and ultraviolet regions (electronic transitions). The characteristic frequencies and corresponding wavenumbers for each of these are given below. Note that we use the relationship c = ln, where l is the wavelength (usually in nm or m) and n is the frequency (s^-1 or Hz). The wavenumber n is inverse of the wavelength in units of cm^-1.

Radiation	Wavelength	Wavenumber	Frequency
Ultraviolet	50–400 nm	25,000–200,000 cm^-1	»10¹⁴–10¹⁵ s^-1
Visible	400–800 nm	12,500–25,000 cm^-1	»10¹⁴ s^-1
Infrared	800 nm–50 m	20–12,500 cm^-1	»10¹⁰–10¹⁴ s^-1
Microwave	50 m–5 cm	0.2–20 cm^-1	»10⁸–10¹⁰ s^-1