Atomic Line Spectra

The Fingerprint of Atoms

Imagine shining white light through a cold gas of hydrogen. Most of the light passes through, but specific colours are missing, appearing as dark lines in the spectrum. Conversely, if you energise the hydrogen gas, it glows with specific bright lines. These lines are not random; they follow a precise mathematical pattern. Each element has a unique set of lines, much like a fingerprint. This uniqueness arises because the energy levels of electrons in different atoms are different. When an electron jumps between these levels, it releases or absorbs a photon with energy exactly equal to the difference between the levels. Since the levels are fixed, the photon energies—and thus the wavelengths—are fixed. This is why we see lines rather than a continuous band of colours.

From Energy Levels to Wavelengths

The energy of a photon is given by \(E = h\nu = \frac{hc}{\lambda}\), where \(h\) is Planck's constant, \(c\) is the speed of light, and \(\lambda\) is the wavelength. In the hydrogen atom, the energy of an electron in level \(n\) is proportional to \(-\frac{1}{n^2}\). When an electron transitions from a higher level \(n_2\) to a lower level \(n_1\), the energy difference is emitted as a photon. The Rydberg formula captures this relationship directly in terms of wavelength. It states that the reciprocal of the wavelength is proportional to the difference of the reciprocals of the squares of the principal quantum numbers.

Here, \(R_H\) is the Rydberg constant for hydrogen, approximately \(1.097\times10^7\,\text{m}^{-1}\). For hydrogen, \(Z=1\). For hydrogen-like ions (ions with only one electron, such as \(\mathrm{He^+}\) or \(\mathrm{Li^{2+}}\)), the nuclear charge \(Z\) scales the energy levels, and thus the wavenumber, by a factor of \(Z^2\). This scaling occurs because the electrostatic attraction between the nucleus and the single electron is stronger for higher \(Z\), making the energy levels more deeply bound.

Spectral Series

Depending on the final energy level \(n_1\) of the transition, the emitted lines group into distinct series. The most famous is the Lyman series, where \(n_1=1\). These transitions involve the ground state and emit ultraviolet light. The Balmer series, with \(n_1=2\), produces visible light and was the first series identified experimentally. The Paschen series (\(n_1=3\)) and Brackett series (\(n_1=4\)) emit infrared radiation. As \(n_2\) approaches infinity, the wavelength of the emitted photon approaches a limit known as the series limit. This limit corresponds to the ionisation energy of the atom from the level \(n_1\). Understanding these series allows us to map the entire electronic structure of the hydrogen atom from a single set of spectral data.

Selection Rules and Intensity

Not all transitions between energy levels are equally probable. Quantum mechanics imposes selection rules that dictate which transitions are "allowed." For electric dipole transitions, the most common type, the change in the principal quantum number \(\Delta n\) can be any integer, but the change in the orbital angular momentum quantum number must be \(\Delta l = \pm 1\). Transitions that violate these rules are "forbidden" and have much lower probabilities, resulting in very weak spectral lines. The intensity of a spectral line depends on the population of the initial state and the transition probability. In thermal equilibrium, lower energy states are more populated, so transitions starting from lower \(n\) levels are generally more intense, provided the transition is allowed. This explains why the \(n=3 \to n=2\) line (H-alpha) is often the brightest in the Balmer series.