Spectral Analysis. While studying the spectrum of a gas...

Spectral Analysis. While studying the spectrum of a gas cloud in space, an astronomer magnifies a spectral line that results from a transition from a $p$ state to an $s$ state. She finds that the line at $575.050 \mathrm{nm}$ has actually split into three lines, with adjacent lines $0.0440 \mathrm{nm}$ apart, indicating that the gas is in an external magnetic field. (Ignore effects due to electron spin.) What is the strength of the external magnetic field?

Key Concepts

Zeeman Effect

The Zeeman effect is the phenomenon where atomic or molecular spectral lines split into multiple components in the presence of an external magnetic field. This splitting occurs because the magnetic field interacts with the magnetic moments associated with the orbital angular momentum (and spin, though it may be ignored in certain cases) of electrons, lifting the degeneracy of energy levels. It is a key tool in astrophysics and atomic physics for diagnosing magnetic fields in various environments.

Normal Zeeman Effect

The normal Zeeman effect refers to the spectral line splitting that occurs when only the orbital angular momentum contributes to the magnetic moment, causing an equally spaced triplet pattern. This occurs under conditions where the electron spin does not play a role, simplifying the analysis by linking the energy level splitting directly to the magnetic field strength through the interaction with the Bohr magneton.

Bohr Magneton

The Bohr magneton is a fundamental physical constant that quantifies the natural unit of the electron's magnetic moment due to its orbital motion. It is used in calculations involving the interaction between an electron’s magnetic moment and an external magnetic field, such as in the context of the Zeeman effect, where the energy shift is directly proportional to the field strength.

Energy-Wavelength Relationship

In spectroscopy, the change in energy between atomic levels is related to the change in wavelength of emitted or absorbed light by the relation ?E = hc/?² · ??, where h is Planck's constant and c is the speed of light. This relationship allows one to convert observed wavelength splittings into energy differences, which can then be used to infer the strength of external perturbations such as magnetic fields.

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