# Is the 21cm line of hyperfine structure hydrogen an allowed transition?

The ground state of hydrogen is $^{2}S_{\frac{1}{2}}$ with nuclear spin $I=\frac{1}{2}$. The levels are split by $F=1,F=0$. By the rules of electric dipole transition, $\Delta l = \pm 1$. But in this case it is going from $l=0$ to $l=0$.

Is this an allowed electric dipole transition?

In the real world transitions are rarely (never?) forbidden because the assumptions we make rarely hold exactly. You are quite correct that the 21 cm transition has $\Delta\ell = 0$ and is therefore forbidden. However it can occur (very slowly) as a magnetic dipole transition.
The lifetime of the excited state is around $10^{15}$ seconds, and we can observe the radiation from the transition only because (a) in interstellar space atoms can go this long between collisions, and (b) there's a hell of a lot of hydrogen out there.
• Also to point out that it is not generally visible in general lab conditions, as the doppler width is about $~\sim 10^{-4}eV$ – user44840 Jun 12 '15 at 16:22