Estimating Atomic Transition Strengths

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Estimating the Strength of Atomic/Molecular Lines

1 Einstein A’s for

is a measure of the probability of decay per unit time, so . 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.