The ground state of sodium \({ }^{2} S_{1 / 2}\) is subject to hyperfine splitting into two levels with a separation of \(1.771 \times 10^{9} \mathrm{~Hz}\).

(a) The atomic structure factor \(\alpha\) introduced in Section 9.6 is \(885.8 \times 10^{6} \mathrm{~Hz}\) for the ground state, as established both experimentally and theoretically. From the data, what is the nuclear spin of the sodium nucleus?

(b) From the result in part (a), what are the \(F\) values of the hyperfine levels in the ground state of sodium?