Estimating Atomic Transition Strengths

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Estimating the Strength of Atomic/Molecular Lines

1 Einstein A’s for

Recall from Einstein Coefficients that is a measure of the probability of decay per unit time, so is approximately the lifetime of an atom in an excited state. This should be about equal to the energy of the electron state divided by the average power radiated by an electron being accelerated:

Now (the electric dipole moment) and for a spring, so:

For H: and so:

2 Magnetic Dipole for

The magnetic dipole of an electron is:

Thus we can estimate the ratio of for magnetic dipole transitions to that of electric dipole transitions:

This tells us that the magnetic dipole states (that is, fine and hyperfine states) are longer lived than electric dipole states by a factor of .

The actual value is .

3 Electric Quadrupole

If one is nearby a rotating quadrupole, one sees the (electric) field rotating rigidly. However, from far away, there are kinks in the field, resulting in a retarded potential. The radiation nearby goes as . For a monopole, the electric field is . For a dipole, it is , where s is the charge separation. For a quadrupole:

Since , the ratio of the powers emitted by a quadrupole vs. a dipole should be:

An acoustic analogy: a kettle whistle is a monopole, a guitar string is a dipole, and a tuning fork (with its two out-of-phase prongs) is a quadrupole.\

Anyway, since ,

Thus , the lowest quadrupole rotational transition of , should have an of about:

The actual value is .

4 Radio Recombination Lines

In HI, the transition has a wavelength of 6 cm. We can estimate its :

This presents the question of which dipole moment to use. It turns out we must use .

5 Back to

Now for doppler broadening, and , so for electric and magnetic dipole transitions:

So the cross-section for these transitions does not depend on wavelength.