The Hyperfine effect is analogous the Zeeman effect with which you should be familiar.  Recall that the Zeeman effect is the splitting of atomic energy states that occurs due to the interaction of the electron's total angular momentum $J$, with an external magnetic field $B$.  In the Hyperfine effect an orbital electron with angular momentum $J$ interacts with the magnetic field produced by the atomic nucleus.  Some nuclei have an intrinsic magnetic dipole moment given by the spin of nucleus $I$.

You will be studying the Hyperfine effect in rubidium (Rb).  Naturally occurring rubidium has two isotopes, ${}^{87}$Rb (28% abundant, nuclear spin $I = 3/2$) and ${}^{85}$Rb (72% abundant, nuclear spin $I = 5/2$). In its ground state, rubidium has a single electron outside closed shells giving it a hydrogen-like spectrum. The interaction of the nuclear spin I and the electron spin J result in a splitting of both the ground and first excited states in Rb.

Ignoring hyperfine effects for the moment, the lowest energy level in which the single electron of rubidium outside the closed shells can reside is the $5^2\mathrm{S}_{1/2}$ state. In this experiment, we will concentrate only on transitions between this state and the next higher energy state, $5^2\mathrm{P}_{3/2}$. Transitions between these states occur via the emission or absorption of a photon of wavelength 780.0 nm. Adding in hyperfine interactions, we find that the $5^2\mathrm{S}_{1/2}$ and $5^2\mathrm{P}_{3/2}$ states split into two and four closely separated states (respectively) which are differentiated by quantum number $F$, where $\vec{F} = \vec{J} + \vec{I}$ is the total atomic angular momentum. As ${}^{87}$Rb and ${}^{85}$Rb have different intrinsic nuclear spins $I = 3/2$ and $I = 5/2$, respectively), the allowed values of $F$ are different for the two isotopes. The energy level diagram for both species of rubidium, including the hyperfine structure, is shown in Figure 1.

Figure 1: Energy level diagrams for ${}^{87}$Rb and ${}^{85}$Rb. (Note that the vertical spacing of the energy levels is not drawn to scale.)

The quantum mechanical selection rules determining the allowed transitions are

$\Delta F = 0, \pm 1$  (but not $0 \rightarrow 0$), $\Delta L = 0, \pm1$, and $\Delta S = 0$. In this experiment you will measure the magnitude of the hyperfine splitting between the $5^2\mathrm{P}_{3/2}$  substates. You will do so by measuring the difference in frequency of photons associated with transitions between the $5^2\mathrm{S}_{1/2}$  states and the hyperfine ($F^\prime$) states. As an example, refer to Fig. 2 which illustrates the energy difference between the $F^\prime = 2$ and $F^\prime = 3$ hyperfine levels of the $5^2\mathrm{P}_{3/2}$  state.

Figure 2: Energy level diagram for ${}^{87}$Rb showing the frequencies of the hyperfine splittings.