Hyperfine splittings are of order .
The first term is spin angular momentum of the nucleus. For S-wave electrons only the last term is nonzero. This is the contact term.
The is 4-fold degenerate with the hyperfine considerations. This
is split into 3 and 1 by the hyperfine structure.
Which corresponds
to
and
. This is really the first excited state of Hyrdorgen.
for l 0 states, the fine structure are coupled
. To calculate
must decouple
,
and recouple
,
. This gives:
.m