Atomic recoil and the polarization of photon emitted

[megadethized]

Hi,

i have a question about the center of mass change during photon emission in terms of the polarization of the emitted photon.

first, a few facts are as follows:

i) selection rule: if delta m=0, that means p (atomic recoil) = p (photon) in magnitude. This is of pi transition and emitted photon is linearly polarized.

ii) selection rule: if delta m=+1 or -1, that means p (atomic recoil) = 0 and emitted photon has +h/2*pi or -h/2*pi momentum. This is of sigma transition (two sigma polarizations) and emitted photons are spherically polarized.

my question is:

in case I, the center of mass (COM) of the atom changes, cause according to simple mechanical theory, we know that "a source emitting a spherical wave cannot recoil, because the spherical symmetry of the wave prevents it from carrying any linear momentum from the source, thus the COM of the atom changes". What about the second case? Due to the existence of atomic recoil, can we say that the COM remains invariant?? in other words, if an atom emits linearly polarized photon after delta m=+1 or -1 transition, the COM remains unchanged? the case I is true, i only wonder about the case 2.

thanks
Berkay

 

[mfb]

p_{atomic recoil} = -p_photon in the rest frame of the initial atom)\ is always true, this has nothing to do with polarizations and the angular (probability) distribution of the emission.
The center of mass does not change.

 

[megadethized]

but i am sure that the COM changes if atom emits spherically polarized photon (i learned this from many textbooks). Cause there is a very well-known Raman-Nath approximation which supplies the neglect of the center of mass change for sigma polarizations (spherically polarized photon emission) and it makes calculations easier. If there was nothing to do with polarizations, Raman-Nath approximation would not be required.

i have only a doubt for the linearly polarized photon emission. In this case, the COM of the atom changes or not? I really need the answer.

 

[DrClaude]

What mfb said is correct. An atom always recoils when emitting a photon; it follows simply from conservation of momentum.

You might be confused by the fact that, for a laser propagating along the z-axis, emission of a $\pi$ photon corresponds to a recoil transverse with respect to the laser (i.e., in the xy-plane), and is de facto ignored in cases where the transverse momentum is not of importance (or considered to be small, due to the lack of stimulated emission).

Also, there is no such thing as "spherically polarized" photons. I guess you mean circularly polarized.

 

[megadethized]

OK, let's suppose that all you have said is correct. But! how does an atom recoil while emitting circularly (or spherically -older terminology-) polarized photon in the existence of spherical symmetry? Do the textbooks tell wrong?

and also, the atomic recoil in Raman-Nath approximation is ignored for sigma photons, not pi one. Anyway, i got my answer, for pi photon emitting, the COM of the atom does not change, OK. Thanks for your help.

 

[DrClaude]

OK, let's suppose that all you have said is correct. But! how does an atom recoil while emitting circularly (or spherically -older terminology-) polarized photon in the existence of spherical symmetry? Do the textbooks tell wrong?

I think there is a communication problem. When you say "spherical", are you talking about the polarization of the photon? I have never seen that word used for polarization. Or are you talking about the distribution of the photons, as mfb pointed out?

What context are you considering? I am not that familiar with the Raman-Nath approximation, other than it has been used for the study of the interaction of an atomic beam with a standing wave from a laser. So, are you talking about a single atom, or the statistical average over many atoms?