# Harvard Astronomy 201b

## Answer to question (3)

In Uncategorized on March 28, 2013 at 7:04 am

In your bathroom, if you clean it—$NH_3$ is actually just ammonia!  Ammonia is common in regions near the galactic center (Kaifu et al. 1975). And CO  becomes optically thick before ammonia—and optically thick is optically useless when you want to find densities!  Ammonia allows study of denser regions—exactly what is needed to probe star formation.

The (1,1) transition is actually quite exotic: no mere rotational line here.  Ammonia is shaped like a triangular pyramid, with the three H’s at the base and the N on top.  The N can quantum mechanically tunnel through the potential barrier of the base, inverting the pyramid.  So the (1,1) transition is also known as an “inversion” transition.

You might wonder why there is a potential barrier at all—after all, each H atom is neutral, and so is the N on top.  But, if you were an electron on one of the H’s, were would you want to be? Certainly far from the other 2 Hs’ electrons!  Hence each H’s electron will spend most of its time outside the base, meaning the triangle formed by the H’s will be slightly positively charged on the inside. Similarly, the electron on the N will want to be as far from the other three electrons as it can be, so it will hover above the N, meaning the bit of the N facing the pyramid’s triangular base will be slightly positively charged.  Ergo, potential barrier.

Making simple assumptions, we can estimate the energy of the inversion.  Assuming the distance to the base’s center for each H’s electron is (a+l), a the Bohr radius and l the ammonia bond length, 1 angstrom, and that the N’s electrons are (a+l) above this center, we calculate the potential where the 7 protons in N are.  Converting to energy and thence frequency, we obtain $\nu\sim 1 \;\rm{GHz}$. Incidentally, to go further one could use the WKB approximation to estimate the tunneling probability.  Given a minimum flux per beam width to which the telescope is sensitive (14 mJy here is the noise), one could even then place a lower bound on the column density observable with this transition by a particular instrument, and assuming isotropy, one could get a density from the column density.