Zeeman splitting

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\begin{document} \title{Zeeman Splitting}

Zeeman splitting is the splitting of energy levels (and from an observational perspective, spectral lines) that occurs when an atom emits in a magnetic field. This effect is easiest to understand for atoms in the singlet state where a semi-classical treatment suffices for explaining splitting, this is called the normal Zeeman effect. For the anomalous Zeeman effect, which occurs in arbitrary spin and angular momentum transitions, a quantum mechanical treatment is necessary.   

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Results from the original experiment conducted by Zeeman. On left is with a magnetic field, at right is without

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\subsection{Normal Zeeman Splitting}
If we consider singlet transitions ($\Delta m = 0, \pm 1$) then the energy change due to magnetic field is, much like we saw when doing fine/hyperfine transitions:

\begin{equation} \Delta E = -\mu \cdot \mathbf{B} = - m_l \mu_e B_{ext} \end{equation} Only here the B field is external. $m_l$ is just the magnetic quantum number (which can range between $-l < m_l < l$ for a given orbital) and $\mu_e = e\hbar/2m_e$. In the simplest case of transition between the $l = 2$ singlet state and the $l = 1$ singlet state which would have given off a photon with frequency $\omega_0$ in the absence of an external field, this gives us $\omega_+ = \omega_0 + eB/2m_e c$, $\omega_0$, and $\omega_- = \omega_0 - eB/2mc$. This rarely happens in nature though, so to understand how to deduce astrophysical magnetic fields from Zeeman splitting we must look a the anomalous Zeeman effect.

\subsection{Anomalous Zeeman Splitting }

Anomalous splitting is the more generic case of Zeeman splitting that arises when our spin is nonzero, meaning that spin orbit coupling occurs and $m_l$ is no longer a good quantum number. In most astrophysical cases of interest the external magnetic field is much lower than the internal magnetic field so we can treat the Zeeman effect as a perturbation to the fine structure field. The perturbing hamiltonian is simply: \begin{equation} H'_z = -(\mu_l +\mu_s) \cdot B_{ext} \end{equation} where $\mu_l$ and $\mu_m$ are the magnetic moments associated with orbital motion and spin, respectively. Since we are only looking at a perturbation, we can still use the 'good' quantum numbers for spin orbit coupling (for more on spin-orbit coupling see Rybicki and Lightman chapter 9), that is, n, l, j and $m_j$. Using that: \begin{equation} \mu_l = -\frac{e}{2m}\mathbf{L} \end{equation} \begin{equation} \mu_m = -\frac{e}{m}{\mathbf{S}} \end{equation} We can redefine our hamiltonian in terms of $\mathbf{S}$ and $\mathbf{L}$ and use first order perturbation theory to find: \begin{equation} \Delta E = \frac{e}{2m} \mathbf{B_{ext}} \cdot \big \langle \mathbf{J} + \mathbf{S} \big \rangle \end{equation} Where we've used that ${\mathbf{J}} = {\mathbf{S}}+{\mathbf{J}}$. Though we know the expectation value of J, we need to find the one for S in this system. We can do this by noting that since \mathbf{J} (total angular momentum) is constant, and $\mathbf{S}$ precesses at high frequency about this vector, the time averaged value for $\mathbf{S}$ is just the projection onto this constant vector: \begin{equation} \mathbf{S_{avg}} = \frac{ \big \langle \mathbf{S} \cdot \mathbf{J}\big \rangle} {\mathbf{J}^2} \mathbf{J} \end{equation} We can then use that $\mathbf{J} = \mathbf{S}+\mathbf{J}$ to find an expression for this dot product in terms of quantities for which we have good quantum numbers: \begin{equation} 2\mathbf{S} \cdot {\mathbf{J} = J^2 + S^2 -L^2 \end{equation} These are all quantities for which we know the eigenvalues! So we can write: \begin{equation} S \cdot J = \frac{1}{2}(J^2 +S^2 - L^2) = \frac{\hbar}{2}[j(j+1) +s(s+1) - l(l+1)] \end{equation} This can be plugged back into our expression for $\mathbf{S_{avg}}$, which can subsequently be plugged back into our expression for $\Delta$ E to give: \begin{equation} \Delta E = \frac{e \hbar B_{ext}}{2m_ec} \left(1+ \frac{j(j+1) +s(s+1) +l(l+1)}{2j(j+1)}\right) m_j = \frac{e \hbar B_{ext}}{2m_ec} g(j, s, l) m_j \end{equation} where g(j,s,l) is the quantity in the parenthesis and is called the Lande g factor. For the purposes of observational astronomy, these gs can all be looked up in a table.

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An illustration of allowed energies due to the Zeeman effect for the l=1 l=2 transition. Note the polarization associated with varying changes in $m_j$. Figure modified from L. Woolsey

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\subsection{Observing the Zeeman Effect}
To get an idea for the change in energy caused by Zeeman splitting we can take a typical magnetic field for the ISM of $10\mu G$ and use the table in Heiles et al. (see references above), to find the shift in the 21 cm hyperfine transition to be on the order of 10 Hz. This is an incredibly small shift and for most astrophysical systems (masers being the exception), is impossible to measure directly. However, we can measure even small shifts due to Zeeman splitting utilizing the polarization of the split lines. Transitions that change values of $l$ but maintain the same value of $m_j$ are $\pi$ polarized where transitions that have a $\Delta m_j$ of $\pm 1$ are $\sigma$ polarized. We can then use that these $\sigma$ polarized lines are elliptically polarized (meaning they are both circular and linearly polarized) to measure the right and left circular polarization of our signal. Adding these two parameters gives the Stokes I parameter where subtracting them gives the Stokes V parameter. If there is no magnetic field we expect our Stokes V parameter to be 0 however, however if there is, the measured V spectrum will be off the form:
\begin{equation}
V(\nu) = \frac{dI (\nu)}{d \nu}bB_{\parallel}
\end{equation}
Where  $B_{\parallel}$ is the line of sight component of the B field and $I(\nu)$ is the Stokes I parameter. This technique makes it possible to measure Zeeman splitting in the ISM.  

\emph{For a review of polarization, see Polarization.}

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