Lyman-alpha line

In physics, the Lyman-alpha line, sometimes written as Ly-$\alpha$ line, is a spectral line of hydrogen, or more generally of one-electron ions, in the Lyman series, emitted when the electron falls from the $n = 2$ orbital to the $n = 1$ orbital, where n is the principal quantum number. In hydrogen, its wavelength of 1215.67 angstroms (121.567 nm or 1.21567 × 10⁻⁷ m), corresponding to a frequency of 2.47 × 10¹⁵ hertz, places the Lyman-alpha line in the vacuum ultraviolet part of the electromagnetic spectrum. Lyman-alpha astronomy must therefore ordinarily be carried out by satellite-borne instruments, except for extremely distant sources whose red-shifts allow the hydrogen line to penetrate the atmosphere.

Because of fine structure perturbations, the Lyman-alpha line splits into a doublet with wavelengths 1215.668 and 1215.674 angstroms. Specifically, because of the electron's spin-orbit interaction, the stationary eigenstates of the perturbed Hamiltonian must be labeled by the total angular momentum j of the electron (spin plus orbital), not just the orbital angular momentum $l$. In the $n = 2$ orbital, there are two possible states, $j = 1/2$ and $j = 3/2$, resulting in a spectral doublet. The $j = 3/2$ state is of higher energy (less negative) and so is energetically farther from the $n = 1$ orbital to which it is transitioning. Thus, the $j = 3/2$ state is associated with the more energetic (shorter wavelength) spectral line in the doublet.

A K-alpha line, or K_α, analogous to the Lyman-alpha line for hydrogen, occurs in the high-energy induced emission spectra of all chemical elements, since it results from the same electron transition as in hydrogen. The equation for the frequency of this line (usually in the X-ray range for heavier elements) uses the same base-frequency as Lyman-alpha, but multiplied by a (Z−1)² factor to account for the differing atomic numbers (Z) of heavier elements, as approximated by Moseley's law.

The Lyman-alpha line is most simply described by the {n,m} = {1,2...} solutions to the empirical Rydberg formula for hydrogen's Lyman spectral series. (The Lyman-alpha frequency is produced by multiplying the Rydberg frequency for the atomic mass of hydrogen, R_M (see Rydberg constant), by a factor of 1/1² − 1/2² = 3/4.) Empirically, the Rydberg equation is in turn modeled by the semi-classical Bohr model of the atom.

See also

2