Atoms in Magnetic Fields

In this section we treat the interaction of the electron's orbital and spin angular momentum with external static magnetic fields. Previously, in the chapter on fine structure, we have considered the spin-orbit interaction: the coupling of electron spin to the magnetic field generated by the nucleus (which appears to move about the electron in the electron's rest frame). The spin orbit interaction causes the orbital and spin angular momenta of the electron to couple together to produce a total spin which then couples to the external field; the magnitude of this coupling is calculated here, first for weak external fields.

The Lande g-factor

Magnetic moment of circulating charge (classical)

The energy of interaction of a classical magnetic moment Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\bf{\mu}} } with a magnetic field { ${\bf {B}}$ } is

$U=-{\bf {\mu }}\cdot {\bf {B}}$

indicating that the torque tends to align the moment along the field. In classical electrodynamics the magnetic moment of a moving point particle about some point in space is independent of the path which it takes, but depends only on the product of the ratio of its charge to mass $m$, and angular momentum $\ell$. This result follows from the definitions of angular momentum

${\bf {L}}\equiv {\bf {r}}\times {\bf {p}}=m[{\bf {r}}\times {\bf {v}}]$

and magnetic moment

${\bf {\mu }}\equiv {\frac {1}{2}}{\bf {r}}\times {\bf {i}}={\frac {q}{2}}[{\bf {r}}\times {\bf {v}}]$

where { ${\bf {i}}$ } is the current and { ${\bf {v}}$ } the velocity (see Jackson Ch.5). The equality of the bracketed terms implies

${\bf {\mu }}={\frac {q}{2m}}{\bf {L}}\equiv \gamma _{\ell }{\bf {L}}$

where $\gamma _{\ell }$ is referred to as the \emph{gyromagnetic ratio}. This is a general result for any turbulently rotating blob provided only that it has a constant ratio of charge to mass throughout. For an electron with orbital angular momentum ${\bf {\ell }}$

${\bf {\mu }}_{\ell }=-{\frac {e}{2m}}{\bf {L}}\equiv -\mu _{B}{\bf {L}}/\hbar$

which is the classical result, and $\mu _{B}$ is the \emph{Bohr magneton}:

Failed to parse (unknown function "\unit"): {\displaystyle \mu_B = \frac{e\hbar}{2m} = \unit{9.27408(4)\times 10^{-24}}{\joule\per\tesla} \rightarrow \unit{1.39983 \times 10^4}{\mega\hertz} \times B/(\text{Tesla}) }

Intrinsic electron spin and magnetic moment

When Uhlenbeck and Goudsmit suggested \cite{Uhlenbeck1926} that the electron had an intrinsic spin $S={\frac {1}{2}}$, it soon became apparent that it had a magnetic moment twice as large as would be expected on the basis of Eq.\ \ref{EQ_magmoment4}. (This implies that the electron cannot be made out of material with a uniform ratio of charge to mass.) This is accounted for by writing for the intrinsic electron moment

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\bf{\mu}} _s = -g_e \mu_B { {\bf{S}} }/\hbar }

where the quantity Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g_e = 2} is called the electron $g$-factor. (The negative sign permits treating Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g_e} as a positive quantity, which is the convention.) This factor was predicted by the Dirac theory of the electron, probably its greatest triumph. Later, experiments by Kusch, followed by Crane et al., and then by Dehmelt and coworkers, have shown (for both electrons and positrons).

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{g_e}{2} = 1.001 159 652 1869(41) }

This result has been calculated from quantum electrodynamics, which gives

${\frac {g_{e}}{2}}=1+{\frac {1}{2}}{\left({\frac {\alpha }{\pi }}\right)}-0.3258{\left({\frac {\alpha }{\pi }}\right)^{2}}+0.13{\left({\frac {\alpha }{\pi }}\right)^{3}}+\cdots$

The agreement betwen the prediction of quantum electrodynamics and experiment on the electron g-factor is often cited as the most precise test of theory in all of physics.

Vector model of the Land\'{e} \texorpdfstring{Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g} }{g}-factor

In zero or weak magnetic field the spin orbit interaction couples { Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\bf{S}}} } and ${\bf {L}}$ together to form Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle { {\bf{J}} } = {\bf{L}} + { {\bf{S}} }} , and this resultant angular momentum interacts with the applied magnetic field with an energy

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U = - g_j \mu_B { {\bf{B}} } \cdot { {\bf{J}} }/\hbar }

which defines $g_{j}$. The interaction of the field is actually with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\bf{\mu_s}} } and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\bf{\mu_{\ell}}} } , however $g_{j}$ is not simply related to these quantities because Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\bf{\mu_s}} } and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\bf{\mu_{\ell}}} } precess about { ${\bf {J}}$ } instead of the field. As Land\'{e} showed in investigations of angular momentum coupling of different electrons \cite{Lande1923}, it is a simple matter to find Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g_j} by calculating the sum of the projections of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\bf{\mu_s}} } and ${\bf {\mu _{\ell }}}$ onto { Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\bf{J}}} }. The projection of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\bf{\mu_{\ell}}} } on { ${\bf {J}}$ } is

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mu_{\ell j} = \frac{-\mu_B |{ {\bf{L}} }|}{\hbar} \frac{ {\bf{L}} \cdot { {\bf{J}} }}{|{ {\bf{L}} |{ {\bf{J}} }}} }

The projection of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\bf{\mu_{s}}} } on { ${\bf {J}}$ } is

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mu_{s j} = - g_e \mu_B \frac{|{{ {\bf{S}} }}|}{\hbar} \frac{{ {\bf{S}} } \cdot { {\bf{J}} }}{ \left| {{\bf{S}}} \right|} { {{\bf{J}}} }| }

The definition of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g_j} gives

$g_{j}=-{\frac {(\mu _{\ell j}+\mu _{sj})}{\left|{\bf {J}}\right|}}\mu _{B}/\hbar$

Taking Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g_e = 2}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} g_j &= \frac{ {\bf{L}} \cdot ( {\bf{L}} + { {\bf{S}} })+ 2 { {\bf{S}} } \cdot ( {\bf{L}} +{ {\bf{S}} })}{ \left| {{\bf{J}}} \right|} ^2 \\ &= 1 + \frac{j(j+1) + s(s+1) - \ell (\ell + 1)}{2j(j+1) } \end{align}}

using $2{\bf {L}}\cdot {\bf {S}}={\bf {J}}^{2}-{\bf {L}}^{2}-{\bf {S}}^{2}$. If a transition from a level with angular momentum Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j^\prime} to a level with $j^{\prime \prime }$ takes place in a magnetic field, the resulting spectral line will be split into three or more components---a phenomenon known as the Zeeman effect. For transitions with a particular $\Delta m$, say $\Delta m=-1$, the components will have shifts

$\Delta E_{z,m,-1}={\left[g_{j^{\prime }}m-g_{j^{\prime \prime }}(m-1)\right]}\mu _{B}B={\left[(g_{j^{\prime }}-g_{j^{\prime \prime }})m-g_{j^{\prime \prime }}\right]}\mu _{B}B$

If $g_{j^{\prime }}=g_{j^{\prime \prime }}$ (or if $j^{\prime }$ or $j^{\prime \prime }=0$) then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta E_{z,m,-1}} will not depend on Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m} (or there will be only one transition with $\Delta m=-1$) and there will be only 3 components of the line (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta m =+1,0 -1} ); this is called the normal Zeeman splitting. If neither of these conditions holds, the line will be split into more than 3 components and the Zeeman structure is termed "anomalous"---it can't be explained with classical atomic models.

Hyperfine structure in an applied field

The Hamiltonian in an applied field Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle { {\bf{B}} }_0 } is

$H=ah{\bf {I}}\cdot {\bf {J}}-{\bf {\mu }}_{J}\cdot {\bf {B}}_{0}-{\bf {\mu }}_{I}\cdot {\bf {B}}_{0}=ah{\bf {I}}\cdot {\bf {J}}+g_{j}\mu _{B}{\bf {J}}\cdot {\bf {B}}_{0}-g_{I}\mu _{B}{\bf {I}}\cdot {\bf {B}}_{0}$

By convention, we take Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\bf{\mu}} _J = - g_j\mu_B { {\bf{J}} }} . Note that we are expressing the nuclear moment in terms of the Bohr magneton, and that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g_I << g_j} . (The nuclear moment is often expressed in terms of the nuclear magneton, in which case ${\bf {\mu }}_{I}$ = Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g_i^\prime \mu_N { {\bf{I}} }} , where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mu_N} is the nuclear magneton.) What are the quantum numbers and energies? Before discussing the general solution, let us look at the limiting cases.

Low field

The total angular momentum is ${\bf {F}}={\bf {I}}+{\bf {J}}$. In low field, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m_F} are good quantum numbers. Each level $F$ contains Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (2F+1)} degenerate states. In a weak field Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle { {\bf{B}} }_0 } the ($2F+1$) fold degeneracy is lifted. We can treat the terms

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H_z = - ( {\bf{\mu}} _j + {\bf{\mu}} _I ) \cdot { {\bf{B}} }_0 }

as a perturbation. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\bf{J}} } and ${\bf {I}}$ are not good quantum numbers, only their components parallel to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\bf{F}} } are important. Thus

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \langle{{ {\bf{J}} }\cdot { {\bf{B}} }_0 }\rangle = \frac{\langle{{ {\bf{J}} }\rangle\cdot { {\bf{F}} }} { {\bf{F}} }\cdot { {\bf{B}} }_0}{F^2} }

$H_{z}=-\mu _{B}{\left[-g_{j}({\bf {J}}\cdot {\bf {F}})+g_{I}({\bf {I}}\cdot {\bf {F}})\right]}{\frac {{\bf {F}}\cdot {\bf {B}}_{0}}{F^{2}}}$

Since Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g_I \ll g_j } , we can usually neglect it. We can rewrite this result as

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H_z = g_F \mu_B mB_0 }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g_F = \frac{ \langle{{ {\bf{J}} }\cdot { {\bf{F}} }}\rangle}{F^2} g_j = \frac{g_j}{2} \frac{F(F+1) + j (j+1) - I (I+1)}{F(F+1)} }

For example, let $I=3/2$, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j= 1/2} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F = 1 \text{ or } 2 } . Then

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} F&=1: & W(1) &= -(5/4) ah & g_F &= -g_j /4 \\ F&=2: & W(2) &= (3/4) ah & g_F& = g_j /4 \end{align}}

\begin{figure} \centering

\caption{Total angular momentum F=I+J. (a) In low field, only the components of J and I parallel to F are important. (b) In high field, $m_{i}$ and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m_j} are good quantum numbers.}

\end{figure}

High field

If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\bf{\mu}} _j\cdot { {\bf{B}} }_0 \gg a h { {\bf{I}} }\cdot { {\bf{J}} }} , then { Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\bf{J}}} } is quantized along ${\bf {B}}_{0}$. Although Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\bf{\mu}} _I\cdot { {\bf{B}} }_0 } is not necessarily large compared to the hyperfine interaction, the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle { {\bf{I}} } \cdot { {\bf{J}} }} coupling assures that { Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\bf{I}}} } is also quantized along ${\bf {B}}_{0}$. Thus Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m_I} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m_j} are good quantum numbers. In this case, Eq.\ \ref{EQ_hsaf1} can be written

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H=ahm_im_j + g_j\mu_B m_j B_0 - g_I \mu_B m_I B_0 }

The second term on the right is largest. Usually the first term is next largest, and the nuclear terms is smallest. Figure \ref{fig:B-field-levels} shows low and high field behavior for hyperfine structure for $I=3/2,J=1/2$. \begin{figure} \centering

\caption{Energy level structure for a single-electron atom with nuclear spin 3/2 in the limits of low and high fields.}

\end{figure}

General solution

Finding eigenfunctions and eigenvalues of the hyperfine Hamiltonian for arbitrary field requires diagonalizing the energy matrix in some suitable representation. To obtain a rough idea of the expected results, one can smoothly connect the energy levels at low and high field, bearing in mind that $m=m_{I}+m_{j}$ is a good quantum number at all fields. For $J=1/2$, the eigenvalues of (Eq.\ \ref{EQ_hsaf1}) can be found exactly. The energies are given by the Breit-Rabi formula

$W(m)=-{\frac {1}{2}}{\frac {\Delta W}{2I+1}}-g_{I}\mu _{B}B_{0}m\pm {\frac {\Delta W}{2}}{\sqrt {1+{\frac {4mx}{2I+1}}+x^{2}}},$

where the $+$ sign is for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F = I + 1/2} , and the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -} sign is for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F = I - 1/2 } . $\Delta W$ is the zero field energy separation.

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta W = W (F=I+ 1/2) - W(F= I - 1/2) = ah {\left(\frac{2I+1}{2}\right)} }

The parameter Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} is given by

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x = \frac{(g_e + g_I) \mu_B B_0}{\Delta W} }

Physically, $x$ is the ratio of the paramagnetic interaction (the "Zeeman energy") to the hyperfine separation. The Breit-Rabi energy level diagram for hydrogen and deuterium are shown in figure \ref{fig:Breit-Rabi}. The units reflect current interest in atom trapping. Low-field quantum numbers are shown. It is left as an exercise to identify the high field quantum numbers. \begin{figure} \centering

\caption{Energy level structure for a single-electron atom with nucleaar spin I = 1/2, such as hydrogen (left), and I = 1, such as deuterium (right). From {\it Molecular Beams} by N.F. Ramsey \cite{Ramsey1956}.}

\end{figure} \putbib