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Unformatted text preview: The 21 cm Line: Hyperfine Structure and Applications to Astrophysics Chris Chronopoulos June 3, 2007 Abstract In this paper, we focus on the quantum mechanical nature of the 21 cm emission line of neutral hydrogen. The cause and characteristics of the emission are derived by exploring the eigenvalue problem for angular momentum and applying first-order perturbation theory. We then discuss how the physical characteristics of the radiation make it particularly appropriate for astronomical and astrophysical observations, and describe the use of the line in determining the differential rotation curve for the Milky Way galaxy. 1 Introduction Hydrogen is by far the most abundant element in the universe. It accounts for about three-quarters of all elemental matter . Furthermore, its internal structure is very well understood, and its energy levels known with great accuracy — indeed, the hy- drogen atom is one of the precious few real-world quantum systems that can be solved for analytically. Experimentally, the Rydberg constant has been confirmed with a rel- ative uncertainty of less that 7 parts per trillion . Because of these facts, hydrogen is a natural target for observational astronomy. Its well-defined spectral lines allow for the determination of both matter distributions and velocity data, especially in galaxies and clusters, where neutral hydrogen is prevalent. It is precisely this reasoning which led Dutch astronomer Jan Oort to ponder the existence of a hydrogen transition line in the radio spectrum (18 MHz - 10 GHz, for all practical astronomical purposes ). Oort had been using optical methods to 2 Chris Chronopoulos study the structure of the Milky Way galaxy for years, only to be hindered by the optical spectrum’s tendency to be attenuated by the large clouds of dust and gas that pervade the galactic disk. Even with the best telescopes, optical astronomers can only see a few thousand light years across the galaxy, which has a diameter of about 80,000 light years . Oort assigned H.C. Van de Hulst, an apprentice of his, the task of finding a transition line in hydrogen that occurs at radio frequencies . Hulst analyzed what is now called hyperfine structure and thus derived the slight energy difference between the spin-up and spin-down states in the electron wave function. Qualitatively, this can be thought of as the transition between the state in which the electron magnetic moment is parallel to that of the proton, and that in which the spins are antiparallel. In the ground state, the energy difference corresponds to a 1420 MHz, or 21 cm, spectral line. 2 Derivation The perturbation to the Coulomb potential of the hydrogen atom arises from the interaction between the spin magnetic moment of the nuclear proton and that of the electron. The perturbing Hamiltonian is the dipole-dipole interaction: H =- ~μ e · ~ B p (1) where ~μ e is the intrinsic magnetic moment of the electron and ~ B p is the magnetic field produced by the proton’s moment. These magnetic moments are different forfield produced by the proton’s moment....
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- Spring '11
- Physics, Redshift, rotation curve, Chris Chronopoulos, hyperﬁne structure