Hyperfine States of Alkali Atoms

© 2006 Krishna Myneni

Introduction

The alkali atoms are those which reside in the first column of the periodic table. These atoms have a single valence electron in the $s$ shell, and their quantum mechanical treatment is effectively that of a single electron atom, e.g. the hydrogen atom. The atomic level structure of the alkali atoms, including various internal interactions, and under external perturbations such as static electric and magnetic fields, may then be treated analytically. Thus, the alkali atoms have been important, historically, in the experimental validation of the predictions of quantum mechanics. Alkali atoms are also important from a technological standpoint. Their ground hyperfine states, which we discuss here, are used in setting the standard for time and frequency measurements. In addition, these states have importance in astrophysics, since much of the information obtained from interstellar clouds is due to radio emissions at 21 cm, emitted in transitions between hyperfine levels of the hydrogen atom.

In this paper we discuss how the various internal angular momenta of an atom couple to give its total angular momentum, and the corresponding angular momentum eigenstates of the atom. The hyperfine levels are associated with these eigenstates. A specific example is worked out for the ground state of the alkali atom, $^{87}$Rb.

Angular Momentum States of an Atom

The total angular momentum of an atom consists of various parts: the orbital angular momentum of each electron, $\vec{\ell_i}$, the intrinsic spin of each electron, $\vec{s_i}$, and the intrinsic spin of the nucleus, $\vec{I}$, which is actually a combined angular momentum of the nucleons, except for hydrogen. Due to interactions inside the atom, the individual momenta are not conserved separately -- only the total angular momentum of the atom is conserved. The total angular momentum is given by[1]

\begin{displaymath} \vec{F} = \vec{J} + \vec{I} = {\sum_i \vec{\ell_i}} + {\sum_i \vec{s_i}} + \vec{I} \end{displaymath} (1)

where $\vec{J}$ is the total electronic angular momentum, consisting of the sum of the individual electronic angular momenta, $\vec{\ell_i}$, plus the sum of the individual electronic spins, $\vec{s_i}$, and $\vec{I}$ is the nuclear spin. It should be noted that the individual angular momenta, $\vec{\ell_i}$, $\vec{s_i}$, and $\vec{I}$ are vector operators, each with an associated quantum number denoting the permitted eigenvalues of the magnitude of the angular momentum. Thus, the operator $\vec{J}$ has an associated quantum number $J$, with the magnitude of the angular momentum being given by $\sqrt{J(J + 1)}\hbar$.

If there were no interaction between the nucleus and the electrons, other than the Coulomb force, the set of quantum numbers $(J, I, m_J, m_I)$ for the operators $J^2= \vec{J}\cdot\vec{J}$ and $I^2$, and operators for the $z$-components of these angular momenta, $J_z$ and $I_z$, would be sufficient and complete for uniquely labeling the angular momentum states of the atom. However, the magnetic dipole field arising from the nuclear spin exerts torque on the electrons, coupling $\vec{J}$ and $\vec{I}$. Only the magnitude of the total angular momentum $\vert\vec{F}\vert$, and its $z$-component, $F_z$, can be considered as strictly conserved quantities. This coupling is known as the hyperfine interaction, because it removes the degeneracy of the $m_J, m_I$ sub-levels and leads to very fine structure in the spectra of atoms. Hyperfine structure in the spectra of atoms can be observed only by very high resolution instruments (large grating spectrometers, Fabry-Perot interferometers, or laser spectrometers).

The hyperfine coupling causes the $z$-components of $\vec{J}$ and $\vec{I}$ to not be conserved. Therefore, the quantum numbers $m_J$ and $m_I$ are not ``good'' quantum numbers, i.e. the eigenstates of $J_z$ and $I_z$ are no longer angular momentum eigenstates of the atom. The new angular momentum eigenstates, in the presence of the hyperfine interaction, are given by the simultaneous eigenstates of the operators $F^2$ and $F_z$. These eigenstates, which give the hyperfine levels of the atom, are labeled by the new set of quantum numbers $(F, m_F)$. In practice, the hyperfine interaction is very weak, and the quantum numbers $J$ and $I$ are still useful to label distinct states of the atom, so the angular momentum eigenstates are denoted by $\vert\ J\ I\ F\ m_F\ \rangle$.

For given $J$ and $I$, the allowed values of the quantum number for the total angular momentum, $F$, are given by the coupling rules for two angular momenta[2][3]:

\begin{displaymath} F =\ \vert J - I\vert,\ \vert J - I\vert+1,\ \ldots,\ J+I-1,\ J+I \end{displaymath} (2)

with $m_F$ varying from $-F,\ -F+1,\ \ldots,\ F-1,\ F+1$. The total number of allowed states remains the same in the coupled and uncoupled representations. The hyperfine interaction removes the degeneracy of the angular momentum states, producing very small energy differences for states having different $F$ quantum numbers.

Hyperfine States

The angular momentum eigenstates of the atom, $\vert\ J\ I\ F\ m_F\ \rangle$, can be written as superpositions of the uncoupled states $\vert\ J\ I\ m_J\ m_I\ \rangle$:

\begin{displaymath} \vert\ F\ m_F\ \rangle = \sum_{m_J, m_I}{c_{J, I, F, m_J, m_I, m_F} \vert\ J\ I\ m_J\ m_I\ \rangle} \end{displaymath} (3)

where, henceforth, the $J$ and $I$ are implied in the coupled-state notation $\vert\ F\ m_F\ \rangle$. The superposition coefficients $c$ are known as the Clebsch-Gordan coefficients, given by
\begin{displaymath} c = \langle\ J\ I\ F\ m_F\ \vert\ J\ I\ m_J\ m_I\ \rangle = ... ...ay}{ccc} J & I & F \\ m_J & m_I & -m_F \end{array}\right) \end{displaymath} (4)

where the matrix term denotes a Wigner $3j$ symbol. For a set of coupled and uncoupled quantum numbers, $(J, I, F, m_J, m_I, m_F)$, the Wigner $3j$ symbol is a number which can either be evaluated from a lengthy formula, or which can be looked up in tables -- the interested reader can see Chapter 5 or Appendix C of reference [1] for both methods.

Since angular momentum conservation requires

\begin{displaymath} m_F = m_J + m_I \end{displaymath}

the double sum in equation 3 can be replaced by a single sum over $m_J$, and using eq 4 we can write
\begin{displaymath} \vert\ F\ m_F\ \rangle = \sum_{m_J = -J}^{+J}{ (-1)^{J - I ... ...m_F \end{array}\right) \vert\ J\ I\ m_J\ (m_F-m_J)\ \rangle } \end{displaymath} (5)

Equations 3-5 are valid in general for any atom in which the angular momenta are coupled as described in eq 1. Thus far, we have not imposed any requirements which are particular to alkali atoms. For the particular case of the ground state of an alkali atom, only the valence $s$-shell electron contributes to $\vec{J}$. It follows that, in this case, the quantum number $J$ is given by

\begin{displaymath} J=\ell + s = 0 + {1\over 2} = {1\over 2} \end{displaymath}

and there are then only two terms in the sum for eq 5. For this particular case, we may write explicitly,

\begin{displaymath} \vert\ F\ m_F\ \rangle = (-1)^{1/2 - I + m_F}\sqrt{2F+1}\lef... ...ft\vert\ -{1\over 2},\ m_F+{1\over 2}\ \right\rangle + \right. \end{displaymath}


\begin{displaymath} \left. \left( \begin{array}{ccc} {1\over 2} & I & F \\ +... ...eft\vert\ +{1\over 2},\ m_F-{1\over 2}\ \right\rangle \right] \end{displaymath} (6)

Example: Ground-State Hyperfine Levels of $^{87}$Rb

For $^{87}$Rb, in the ground $S_{1/2}$ state, $I = 3/2$ and $J = 1/2$. Therefore,

\begin{displaymath} m_I = -{3\over 2}, -{1\over 2}, +{1\over 2}, +{3\over 2} \end{displaymath}

and

\begin{displaymath} m_J = -{1\over 2}, +{1\over 2} \end{displaymath}

giving 8 possible uncoupled angular momentum basis states $\vert\ J\ I\ m_J\ m_I\ \rangle$. These states are represented compactly below using the notation $\vert n\rangle$, where $n = 1, 2,\ldots, 8$. The values of $J$ and $I$ have been omitted for clarity below, since they are fixed.
$\displaystyle \vert 1 \rangle$ $\textstyle \equiv$ $\displaystyle \left\vert\ m_J = -{1 \over 2},\ m_I = -{3 \over 2}\ \right\rangle$ (7)
$\displaystyle \vert 2 \rangle$ $\textstyle \equiv$ $\displaystyle \left\vert\ +{1 \over 2},\ -{3 \over 2}\ \right\rangle$ (8)
$\displaystyle \vert 3 \rangle$ $\textstyle \equiv$ $\displaystyle \left\vert\ -{1 \over 2},\ -{1 \over 2}\ \right\rangle$ (9)
$\displaystyle \vert 4 \rangle$ $\textstyle \equiv$ $\displaystyle \left\vert\ +{1 \over 2},\ -{1 \over 2}\ \right\rangle$ (10)
$\displaystyle \vert 5 \rangle$ $\textstyle \equiv$ $\displaystyle \left\vert\ -{1 \over 2},\ +{1 \over 2}\ \right\rangle$ (11)
$\displaystyle \vert 6 \rangle$ $\textstyle \equiv$ $\displaystyle \left\vert\ +{1 \over 2},\ +{1 \over 2}\ \right\rangle$ (12)
$\displaystyle \vert 7 \rangle$ $\textstyle \equiv$ $\displaystyle \left\vert\ -{1 \over 2},\ +{3 \over 2}\ \right\rangle$ (13)
$\displaystyle \vert 8 \rangle$ $\textstyle \equiv$ $\displaystyle \left\vert\ +{1 \over 2},\ +{3 \over 2}\ \right\rangle$ (14)

Using eq 6 with $I = 3/2$, consulting a table of $3j$ symbols[1], and, from eq 2, noting that the allowed values of $F$ are

\begin{displaymath} F = \left\vert{3\over 2} - {1\over 2}\right\vert,\ \left({3\over 2} + {1\over 2}\right)\ =\ 1, 2 \end{displaymath}

we obtain the coupled hyperfine states $\vert\ F\ m_F\ \rangle$,
$\displaystyle \vert\ 1,\ -1\ \rangle$ $\textstyle =$ $\displaystyle \sqrt{3 \over 4}\ \vert 2\rangle - {1\over 2}\ \vert 3\rangle$ (15)
$\displaystyle \vert\ 1,\ \ 0\ \rangle$ $\textstyle =$ $\displaystyle \sqrt{1 \over 2}\ \vert 4\rangle - \sqrt{1\over 2}\ \vert 5\rangle$ (16)
$\displaystyle \vert\ 1,\ +1\ \rangle$ $\textstyle =$ $\displaystyle {1 \over 2}\ \vert 6\rangle - \sqrt{3\over 4}\ \vert 7\rangle$ (17)
$\displaystyle \vert\ 2,\ -2\ \rangle$ $\textstyle =$ $\displaystyle \vert 1 \rangle$ (18)
$\displaystyle \vert\ 2,\ -1\ \rangle$ $\textstyle =$ $\displaystyle {1 \over 2}\ \vert 2\rangle + \sqrt{3\over 4}\ \vert 3\rangle$ (19)
$\displaystyle \vert\ 2,\ \ 0\ \rangle$ $\textstyle =$ $\displaystyle \sqrt{1 \over 2}\ \vert 4\rangle + \sqrt{1\over 2}\ \vert 5\rangle$ (20)
$\displaystyle \vert\ 2,\ +1\ \rangle$ $\textstyle =$ $\displaystyle \sqrt{3 \over 4}\ \vert 6\rangle + {1\over 2}\ \vert 7\rangle$ (21)
$\displaystyle \vert\ 2,\ +2\ \rangle$ $\textstyle =$ $\displaystyle \vert 8 \rangle$ (22)

For $^{87}$Rb, the difference in energy between the $F=1$ and $F=2$ hyperfine levels is $\Delta E = h\Delta \nu$, with $\Delta \nu = 6.8347$ GHz[4]. The three sublevels for $F=1$: $\vert 1, -1\rangle$, $\vert 1, 0\rangle$, and $\vert 1, -1\rangle$ are degenerate, as are the five sublevels for $F=2$. The degeneracy of the sublevels can be removed by placing the atom in an external magnetic field.

Bibliography

1
R. D. Cowan, The Theory of Atomic Structure and Spectra, (University of California Press, 1981).

2
S. Gasiorowicz, Quantum Physics, 3$^{rd}$ ed., (Wiley and Sons, 2003).

3
E. Merzbacher, Quantum Mechanics, 3$^{rd}$ ed., (Wiley and Sons, 1998).

4
D. Steck, Rubidium 87 D Line Data, rev 1.6, http://steck.us/alkalidata (2003).